On the (non-)contractibility of the order complex of the coset poset of a classical group

Let G be a classical group and suppose that G does not contains non-trivial graph automorphisms. In this paper we prove that the order complex of the coset poset of G is non-contractible. In order to prove it, we show that PG(−1) does not vanish, where PG(s) is the Dirichlet polynomial associated to the group G.