David J. Green

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The Cohomology of the Groups of Order 128

In total, there are 2328 groups of order 128. The first complete cohomology computation in summer 2008 was done using Sage, versions 3.0.6 up to 3.1.1, parallely on 2 Dual Core AMD Opteron(tm) Processor 270 (2 GHz), with an earlier version of our package. In this setting, the cohomology of all but four groups was obtained after a total of roughly 3 weeks. The four most difficult cases (2298, 2300, 2326, 2327) were eventually finished after very roughly one additional month for each example.

In the meantime, our package underwent a considerable improvement. In order to test the new performance, we made a second complete computation in May 2009, on the central Sage server, a 24-core Sun X4450 with 2.66 GHz and 128 GB RAM. However, we never had more than four parallel processes. We are grateful that William Stein gave us the opportunity to work on that machine. We acknowledge the support by National Science Foundation Grant No. DMS-0821725.

In this setting, we found:

• Only one example required more than 2 GB of RAM (namely 3.6 GB).
• The total computation time for all but three examples was about one week, with at most four parallel computations (often only three).
• The cohomology computation for group number 2298 and for group number 2300 took about one week each.
• The cohomology computation for group number 2327 (extraspecial of type minus) took about 11 days.

Groups numbered according to the Small Groups library

Groups with a custom name

Central product SD_16 * SD_16

Dihedral group of order 128

Direct product 64gp32 x C_2

Direct product 64gp138 x C_2

Direct product D8 x V_16

Direct product Q8 x V_16

Extraspecial 2-group of order 128 and type +

Extraspecial 2-group of order 128 and type -

Modular group of order 128

Quaternion group of order 128

Semidihedral group of order 128

Sylow 2-subgroup of 2.PGU_2(31)

Sylow 2-subgroup of exceptional group G_2(3):2

Sylow 2-subgroup of Hall-Janko Group J_2

Sylow 2-subgroup of Mathieu Group M_22

Sylow 2-subgroup of one double cover of Sz(8)

Sylow 2-subgroup of Symmetric Group S_8

Sylow 2-subgroup of Symplectic Group Sp_4(3)

Sylow 2-subgroup of U_3(7)

Wreath product V_8 wr C_2