G = Syl2(S4(4)) is Sylow 2-group of Symplectic group S4(4)
G has 4 minimal generators, rank 6 and exponent 4. The centre has rank 4.
There are 2 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 6, 6.
At present no information on the cohomology ring.