G = Syl2(M22) is Sylow 2-subgroup of M_22
G has 3 minimal generators, rank 4 and exponent 8. The centre has rank 1.
The 7 maximal subgroups are: 64gp136, 64gp138 (2x), Syl2(L3(4)), 64gp32 (2x), 64gp35.
There are 4 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3, 4, 4.
At present no information on the cohomology ring.