Small group number 8 of order 625

G is the group 625gp8

G has 2 minimal generators, rank 3 and exponent 25. The centre has rank 1.

The 6 maximal subgroups are: M125 (5x), V125.

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 20 generators:

There are 151 minimal relations:

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

Essential ideal: There are 5 minimal generators:

Nilradical: There are 16 minimal generators:

Completion information

This cohomology ring was obtained from a calculation out to degree 24. The cohomology ring approximation is stable from degree 24 onwards, and Carlson's tests detect stability from degree 24 onwards.

This cohomology ring has dimension 3 and depth 1. Here is a homogeneous system of parameters:

The first term h1 forms a regular sequence of maximum length. The remaining 2 terms h2, h3 are all annihilated by the class y2.w2 + 2y2.w1.

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.

The ideal of essential classes is free of rank 6 as a module over the polynomial algebra on h1. These free generators are:

The essential ideal squares to zero.

Koszul information

A basis for R/(h1, h2, h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 16.

A basis for AnnR/(h1, h2)(h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 12.

A basis for AnnR/(h1)(h2, h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 10.

A basis for AnnR/(h1)(h2) / h3 AnnR/(h1)(h2) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 14.

Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is V125

Restriction to maximal subgroup number 2, which is 125gp4

Restriction to maximal subgroup number 3, which is 125gp4

Restriction to maximal subgroup number 4, which is 125gp4

Restriction to maximal subgroup number 5, which is 125gp4

Restriction to maximal subgroup number 6, which is 125gp4

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V125

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C5

Poincaré series

(1 + 2t + 2t2 + 2t3 + t4 + t9 + 2t10 + 2t11 + 2t12 + t13) / (1 - t2) (1 - t4) (1 - t10)

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