The Cohomology of the Sylow 2‐Subgroup of J 2

The Hall‐Janko‐Wales group J 2 is one of the twenty‐six sporadic finite simple groups. The cohomology of its Sylow 2‐subgroup S J is computed, an important step in calculating the mod 2 cohomology of J 2 . The spectral sequence corresponding to the central extension for S J is described and shown to collapse at the eighth page. The group S J contains two subgroups 2 _ 1 + 4(the central product of a dihedral and a quaternionic group) and 2 2+4 (the Sylow 2‐subgroup of the matrix group PSL 3 ( 4 )) which detect the cohomology of S J . The cohomology relations for the subgroup 2 2+4 are computed.