Let P be the extraspecial p–group of order p^{2n+1}, of p–rank n + 1, and of exponent p
if p > 2. Let Z be the center of P and let κ_{n,r} be the characteristic classes of degree
2^{n} − 2^{r} (resp. 2(p^{n} − p^{r})) for p = 2 (resp. p > 2), 0 ≤ r ≤ n − 1, of a degree p^{n}
faithful irreducible representation of P. It is known that, modulo nilradical, the ιth
powers of the κ_{n,r}’s belong to T = Im(H^{∗}(P∕Z,F_{p})∕H^{∗}(P,F_{p})∕), with
ι = 1 if p = 2, ι = p if p > 2. We obtain formulae in H^{∗}(P,F_{p})∕ relating the κ_{n,r}^{ι}
terms to the ones of fewer variables. For p > 2 and for a given sequence r_{0},…,r_{n−1} of
nonnegative integers, we also prove that, modulonilradical, the element ∏
_{ri}κ_{n,i}^{ri}
belongs to T if and only if either r_{0} ≥ 2, or all the r_{i} are multiple of p. This gives the
determination of the subring of invariants of the symplectic group Sp_{2n}(F_{p}) in
T .
