  The cohomology ring of a finite group, with coefficients in a finite field,
can be computed by a machine, as Carlson has showed. Here "compute" means to
find a presentation in terms of generators and relations, and involves only the
underlying (graded) ring. We propose a method to determine some of the extra
structure: namely, Stiefel-Whitney classes and Steenrod operations. The
calculations are explicitly carried out for about one hundred groups (the
results can be consulted on the Internet).
  Next, we give an application: thanks to the new information gathered, we can
in many cases determine which cohomology classes are supported by algebraic
varieties.
