G is the group 243gp39
G has 3 minimal generators, rank 3 and exponent 9. The centre has rank 2.
The 13 maximal subgroups are: Ab(9,3,3), E27xC3, M27xC3 (3x), 81gp3 (8x).
There are 4 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3, 3, 3.
At present no information on the cohomology ring.