Group Syl2HS of order 512

G = Syl2HS is Sylow 2-subgroup of the Higman-Sims group HS

G has 3 minimal generators, rank 4 and exponent 8. The centre has rank 1.

There are 9 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3, 3, 3, 3, 3, 4, 4, 4.

This cohomology ring calculation is complete.

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

Ring structure

The cohomology ring has 17 generators:

There are 79 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 17. The cohomology ring approximation is stable from degree 14 onwards, and Benson's tests detect stability from degree 17 onwards.

This cohomology ring has dimension 4 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 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, h3, h4) is as follows.

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

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

Restriction information