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.