Small group number 2222 of order 128

G is the group 128gp2222

G has 5 minimal generators, rank 5 and exponent 4. The centre has rank 2.

There are 3 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 4, 4, 5.

This cohomology ring calculation is complete.

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

Ring structure

The cohomology ring has 13 generators:

There are 38 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 12 onwards, and Benson's tests detect stability from degree 12 onwards.

This cohomology ring has dimension 5 and depth 3. Here is a homogeneous system of parameters:

The first 3 terms h1, h2, h3 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, h4, h5) is as follows.

A basis for AnnR/(h1, h2, h3, h4)(h5) is as follows.

A basis for AnnR/(h1, h2, h3)(h4) is as follows.

Restriction information