The essential ideal in group cohomology does not square to zero

Let G be the Sylow 2-subgroup of the unitary group SU3(4). We find two essential classes in the mod 2 cohomology ring of G whose product is nonzero. In fact, the product is the “last survivor” of Benson–Carlson duality. Recent work of Pakianathan and Yalçın then implies a result about connected graphs with an action of G. Also, there exist essential classes which cannot be written as sums of transfers from proper subgroups. This phenomenon was first observed on the computer. The argument given here uses the elegant calculation by J. Clark, with minor corrections.