  Let G be the Sylow 2-subgroup of the unitary group $SU_3(4)$. 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 Yalcin 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.
