  Let p be an odd prime and let P be a p-group. We examine the order complex of
the poset of elementary abelian subgroups of P having order at least p^2. S.
Bouc and J. Th\'evenaz showed that this complex has the homotopy type of a
wedge of spheres. We show that, for each nonnegative integer l, the number of
spheres of dimension l in this wedge is controlled by the number of
extraspecial subgroups X of P having order p^{2l+3} and satisfying
Omega_1(C_P(X))=Z(X). We go on to provide a negative answer to a question
raised by Bouc and Th\'evenaz concerning restrictions on the homology groups of
the given complex.
