  We describe a new source of counterexamples to the so-called integral Hodge
and integral Tate conjectures. As in the other known counterexamples to the
integral Tate conjecture over finite fields, ours are approximations of the
classifying space of some group BG. Unlike the other examples, we find groups
of type A_n, our proof relies heavily on representation theory, and Milnor's
operations vanish on the classes we construct.
