The Cohomology of the Sylow 2-subgroup of the Higman-sims Group

Recently there has been substantial progress towards computing the mod 2 cohomology of low rank sporadic simple groups. In fact the mod 2 cohomology of every sporadic simple group not containing (Z/2) has been computed, with the notable exceptions of HS (the Higman–Sims group) and Co3 (one of the Conway groups). The reason for these exceptions is that these two groups have large and complicated Sylow 2-subgroups, with many conjugacy classes of maximal elementary abelian subgroups. The Higman–Sims group has order 44, 352, 000 = 2·3·5·7·11, and its largest elementary abelian 2-subgroup is of rank equal to four. It is a subgroup of Co3 of index equal to 11178 = 2 · 3 5 · 23, hence Syl2(HS) is an index 2 subgroup of Syl2(Co3). In this paper we compute the mod 2 cohomology ring of S = Syl2(HS), obtaining explicit generators and relations. In a sequel we will determine the necessary stability conditions for computing the cohomology of HS itself. This is a step towards obtaining a calculation of the mod 2 cohomology of Co3. This latter group is of particular interest because of its relation to a homotopy–theoretic construction due to Dwyer and Wilkerson (see [2]) and it would seem that the most viable way of accessing this group is via HS. The calculation we present is long and highly technical, involving techniques from topology, representation theory and computer algebra. Our main result is the following