  The cohomology of the degree-$n$ general linear group over a finite field of
characteristic $p$, with coefficients also in characteristic $p$, remains
poorly understood. For example, the lowest degree previously known to contain
nontrivial elements is exponential in $n$. In this paper, we introduce a new
system of characteristic classes for representations over finite fields, and
use it to construct a wealth of explicit nontrivial elements in these
cohomology groups. In particular we obtain nontrivial elements in degrees
linear in $n$. We also construct nontrivial elements in the mod $p$ homology
and cohomology of the automorphism groups of free groups, and the general
linear groups over the integers. These elements reside in the unstable range
where the homology and cohomology remain poorly understood.
