Small group number 225 of order 64

G is the group 64gp225

The Hall-Senior number of this group is 174.

G has 4 minimal generators, rank 3 and exponent 4. The centre has rank 2.

There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 3.

This cohomology ring calculation is complete.

Ring structure | Completion information | Koszul information | Restriction information | Poincaré series

Ring structure

The cohomology ring has 7 generators:

There are 6 minimal relations:

A minimal Gröbner basis for the relations ideal consists of this minimal generating set, together with the following redundant relations:

Completion information

This cohomology ring was obtained from a calculation out to degree 12. The cohomology ring approximation is stable from degree 8 onwards, and Benson's tests detect stability from degree 8 onwards.

This cohomology ring has dimension 3 and depth 2. Here is a homogeneous system of parameters:

The first 2 terms h1, h2 form a regular sequence of maximum length.

The first 2 terms h1, h2 form a complete Duflot regular sequence. That is, their restrictions to the greatest central elementary abelian subgroup form a regular sequence of maximal length.

Data for Benson's test:

Koszul information

A basis for R/(h1, h2, h3) is as follows.

A basis for AnnR/(h1, h2)(h3) is as follows.

Restriction information