Small group number 3 of order 125

G = E125 is Extraspecial 5-group of order 125 and exponent 5

G has 2 minimal generators, rank 2 and exponent 5. The centre has rank 1.

There are 6 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 2, 2, 2, 2, 2, 2.

This cohomology ring calculation is complete.

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

Ring structure

The cohomology ring has 12 generators:

There are 50 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 18. The cohomology ring approximation is stable from degree 18 onwards, and Benson's tests detect stability from degree 18 onwards.

This cohomology ring has dimension 2 and depth 1. Here is a homogeneous system of parameters:

The first term h1 forms a regular sequence of maximum length.

The first term h1 forms a complete Duflot regular sequence. That is, its restriction to the greatest central elementary abelian subgroup forms a regular sequence of maximal length.

Data for Benson's test:

Koszul information

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

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

Restriction information