The Cohomology of the Sylow 2-Subgroup of J 2

Abstract

The Hall-Janko-Wales group J2 is one of the twenty-six sporadic finite simple groups. The cohomology of its Sylow 2-subgroup SJ is computed, an important step in calculating the mod 2 cohomology of J2. The spectral sequence corresponding to the central extension for SJ is described and shown to collapse at the eighth page. The group SJ contains two subgroups 2 _ 1 + 4 (the central product of a dihedral and a quaternionic group) and 22+4 (the Sylow 2-subgroup of the matrix group PSL3(𝔽4)) which detect the cohomology of SJ. The cohomology relations for the subgroup 22+4 are computed.