G = E32+ is Extraspecial 2-group of order 32 and type +
The Hall-Senior number of this group is 42.
G has 4 minimal generators, rank 3 and exponent 4. The centre has rank 1.
There are 6 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3, 3, 3, 3, 3.
This cohomology ring calculation is complete.
Ring structure | Completion information | Koszul information | Restriction information | Poincaré series
The cohomology ring has 5 generators:
There are 2 minimal relations:
A minimal Gröbner basis for the relations ideal consists of this minimal generating set, together with the following redundant relation:
This cohomology ring was obtained from a calculation out to degree 8. The cohomology ring approximation is stable from degree 4 onwards, and Benson's tests detect stability from degree 6 onwards.
This cohomology ring has dimension 3 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 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:
A basis for R/(h1, h2, h3) is as follows.
(1 + 4t + 7t2 + 7t3 + 4t4 + t5) / (1 - t2)2 (1 - t4)