Truncated Quillen complexes of p-groups

Let be an odd prime and let be a -group. We examine the order complex of the poset of elementary abelian subgroups of having order at least . Bouc and Th,venaz showed that this complex has the homotopy type of a wedge of spheres. We show that, for each nonnegative integer , the number of spheres of dimension in this wedge is controlled by the number of extraspecial subgroups of having order and satisfying . We go on to provide a negative answer to a question raised by Bouc and Th,venaz concerning restrictions on the homology groups of the given complex.