G is the group 128gp52
G has 2 minimal generators, rank 4 and exponent 16. The centre has rank 1.
The 3 maximal subgroups are: 64gp30 (2x), 64gp88.
There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 4.
At present no information on the cohomology ring.