CHERN CLASSES AND EXTRASPECIAL GROUPS OF ORDER p

A presentation is obtained for the Chern subring modulo nilradical of both extraspecial p-groups of order p5, for p an odd prime. Moreover, it is proved that, for every extraspecial p-group of exponent p, the top Chern classes of the irreducible representations do not generate the Chern subring modulo nilradical. Finally, a related question about symplectic invariants is discussed, and solved for Sp4(Fp). The main innovation in this work is to consider extraspecial groups as central products, and to partition the maximal elementary abelian subgroups of the central product into those which lift to abelian subgroups of the corresponding direct product, and those which do not.