Nilpotency degree of integral cohomology classes of p-groups

Let p be a prime number and let G be a p-group which is not elementary abelian. For every integral cohomology class of G which restricts trivially to all proper subgroups, we show that xi(p) = 0 if p > 2 or deg(xi) is even, and xi(3) = 0 if p = 2 and deg(xi) is odd. This result is applied to get an upper bound, which is \G\/p, for the nilpotence degrees of nilpotent integral cohomology classes of G.