The cohomology of the groups of order 32

Abstract

We have calculated the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mod hyphen 2"> <mml:semantics> <mml:mrow> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mtext>-</mml:mtext> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\bmod \text {-}2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> cohomology rings of all the groups of 32 elements. This paper describes the methods of calculation; the computer routines used can be adapted to assist in the calculation of the modular cohomology rings of other finite groups. We also describe the results of the calculations; the data we have collected provide a substantial increase in the supply of completed calculations in group cohomology, and so we take this opportunity to compare known results and open conjectures.