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 p². 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 Ω₁(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.