Truncated Quillen complexes of p-groups

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) . Bouc and Thévenaz 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 Thévenaz concerning restrictions on the homology groups of the given complex.