The Cohomology of Finite p-Groups

We present here the cohomology rings of all 2-, 3-, 5- and 7-groups of order at most 128, of all but 6 groups of order 243, and of some sporadic examples of order up to 1024. Please see Simon King's collection of cohomology rings of arbitrary finite groups as well.

For each cohomology ring, we provide dimension and depth, the Poincaré series, the a-invariants, a minimal generating set and minimal algebraic relations between the generators.

We also provide some background information on our computational approach. A more detailed account of the underlying theory is provided in our preprint that is published in J. of Algebra 325 (2011), pp. 352-363.

The results were obtained with the optional Sage package p_group_cohomology (Documentation and Installation).

The package currently is in a transitional phase. We provide here the sources of modular_resolution. Together with Cython and Python wrappers in the Sage library, it will soon be a replacement of the old package.

2-Groups 3-Groups 5-Groups 7-Groups
Order 2
Order 4
Order 8
Order 16
Order 32
Order 64
Order 128
Order 256
Order 512
Order 1024
Order 3
Order 9
Order 27
Order 81
Order 243
Order 729
Order 5
Order 25
Order 125
Order 625
Order 7
Order 49
Order 343


The work that we are presenting here was supported by the German Science Foundation (DFG), project numbers GR 1585/4–1 and GR 1585/6–1, and a Marie Curie project MTKD-CT-2006-042685.

Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany

E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 10/05/2015