The AutoKron can't determine the cohomology of $\text{Gr}_5(\mathbb{R}^{10,1})$.
There are 16 possibilities.
Here are their Poincaré polynomials:
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + 2 x^{17} y^{4} + 12 x^{17} y^{3} + x^{16} y^{4} + 14 x^{16} y^{3} + x^{16} y^{2} + 16 x^{15} y^{3} + 2 x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + 2 x^{10} y^{3} + 16 x^{10} y^{2} + x^{9} y^{3} + 14 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + 2 x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + x^{17} y^{4} + 13 x^{17} y^{3} + x^{16} y^{4} + 15 x^{16} y^{3} + 16 x^{15} y^{3} + 2 x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + 2 x^{10} y^{3} + 16 x^{10} y^{2} + x^{9} y^{3} + 14 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + 2 x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + 2 x^{17} y^{4} + 12 x^{17} y^{3} + 15 x^{16} y^{3} + x^{16} y^{2} + 17 x^{15} y^{3} + x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + 2 x^{10} y^{3} + 16 x^{10} y^{2} + x^{9} y^{3} + 14 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + 2 x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + x^{17} y^{4} + 13 x^{17} y^{3} + 16 x^{16} y^{3} + 17 x^{15} y^{3} + x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + 2 x^{10} y^{3} + 16 x^{10} y^{2} + x^{9} y^{3} + 14 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + 2 x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + 2 x^{17} y^{4} + 12 x^{17} y^{3} + x^{16} y^{4} + 14 x^{16} y^{3} + x^{16} y^{2} + 16 x^{15} y^{3} + 2 x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + x^{10} y^{3} + 17 x^{10} y^{2} + x^{9} y^{3} + 15 x^{9} y^{2} + 12 x^{8} y^{2} + 2 x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + x^{17} y^{4} + 13 x^{17} y^{3} + x^{16} y^{4} + 15 x^{16} y^{3} + 16 x^{15} y^{3} + 2 x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + x^{10} y^{3} + 17 x^{10} y^{2} + x^{9} y^{3} + 15 x^{9} y^{2} + 12 x^{8} y^{2} + 2 x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + 2 x^{17} y^{4} + 12 x^{17} y^{3} + 15 x^{16} y^{3} + x^{16} y^{2} + 17 x^{15} y^{3} + x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + x^{10} y^{3} + 17 x^{10} y^{2} + x^{9} y^{3} + 15 x^{9} y^{2} + 12 x^{8} y^{2} + 2 x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + x^{17} y^{4} + 13 x^{17} y^{3} + 16 x^{16} y^{3} + 17 x^{15} y^{3} + x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + x^{10} y^{3} + 17 x^{10} y^{2} + x^{9} y^{3} + 15 x^{9} y^{2} + 12 x^{8} y^{2} + 2 x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + 2 x^{17} y^{4} + 12 x^{17} y^{3} + x^{16} y^{4} + 14 x^{16} y^{3} + x^{16} y^{2} + 16 x^{15} y^{3} + 2 x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + 2 x^{10} y^{3} + 16 x^{10} y^{2} + 15 x^{9} y^{2} + x^{9} y + 13 x^{8} y^{2} + x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + x^{17} y^{4} + 13 x^{17} y^{3} + x^{16} y^{4} + 15 x^{16} y^{3} + 16 x^{15} y^{3} + 2 x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + 2 x^{10} y^{3} + 16 x^{10} y^{2} + 15 x^{9} y^{2} + x^{9} y + 13 x^{8} y^{2} + x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + 2 x^{17} y^{4} + 12 x^{17} y^{3} + 15 x^{16} y^{3} + x^{16} y^{2} + 17 x^{15} y^{3} + x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + 2 x^{10} y^{3} + 16 x^{10} y^{2} + 15 x^{9} y^{2} + x^{9} y + 13 x^{8} y^{2} + x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + x^{17} y^{4} + 13 x^{17} y^{3} + 16 x^{16} y^{3} + 17 x^{15} y^{3} + x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + 2 x^{10} y^{3} + 16 x^{10} y^{2} + 15 x^{9} y^{2} + x^{9} y + 13 x^{8} y^{2} + x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + 2 x^{17} y^{4} + 12 x^{17} y^{3} + x^{16} y^{4} + 14 x^{16} y^{3} + x^{16} y^{2} + 16 x^{15} y^{3} + 2 x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + x^{10} y^{3} + 17 x^{10} y^{2} + 16 x^{9} y^{2} + 13 x^{8} y^{2} + x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + x^{17} y^{4} + 13 x^{17} y^{3} + x^{16} y^{4} + 15 x^{16} y^{3} + 16 x^{15} y^{3} + 2 x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + x^{10} y^{3} + 17 x^{10} y^{2} + 16 x^{9} y^{2} + 13 x^{8} y^{2} + x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + 2 x^{17} y^{4} + 12 x^{17} y^{3} + 15 x^{16} y^{3} + x^{16} y^{2} + 17 x^{15} y^{3} + x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + x^{10} y^{3} + 17 x^{10} y^{2} + 16 x^{9} y^{2} + 13 x^{8} y^{2} + x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{25} y^{5} + x^{24} y^{4} + 2 x^{23} y^{4} + 3 x^{22} y^{4} + 4 x^{21} y^{4} + x^{21} y^{3} + 5 x^{20} y^{4} + 2 x^{20} y^{3} + 4 x^{19} y^{4} + 5 x^{19} y^{3} + 3 x^{18} y^{4} + 8 x^{18} y^{3} + x^{17} y^{4} + 13 x^{17} y^{3} + 16 x^{16} y^{3} + 17 x^{15} y^{3} + x^{15} y^{2} + 14 x^{14} y^{3} + 5 x^{14} y^{2} + 12 x^{13} y^{3} + 8 x^{13} y^{2} + 8 x^{12} y^{3} + 12 x^{12} y^{2} + 5 x^{11} y^{3} + 14 x^{11} y^{2} + x^{10} y^{3} + 17 x^{10} y^{2} + 16 x^{9} y^{2} + 13 x^{8} y^{2} + x^{8} y + 8 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 5 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
Here are the corresponding generator grids to these 16 possibilities: