The $K(n)$-Euler characteristic of extraspecial $p$-groups.

Let p be an odd prime, and let K(n)* denote the nth Morava K-theory at the prime p; we compute the K(n)-Euler characteristic \chi_{n;p}(G) of the classifying space of an extraspecial p-group G. Equivalently, we get the number of conjugacy classes of commuting n-tuples in the group G. We obtain this result by examining the lattice of isotropic subspaces of an even-dimensional Fp-vector space with respect to a non-degenerate alternating form B.