Coset complexes of $p$-subgroups in finite groups

Let $G$ be a finite group and $p$ be a prime. We denote by $C_p(G)$ the poset of all cosets of $p$-subgroups of $G$. We characterize the homotopy type of the geometric realization $\mathrm{\mid }\mathrm{\Delta }{C}_p(G)\mathrm{\mid }$ for $p$-closed groups $G$, which is motivated by K.S.Brown’s Question. We will further demonstrate that $\chi (C_p(G))\equiv \mathrm{\mid }G{\mathrm{\mid }}_{p^{\mathrm{\prime }}}(\text{mod}p)$ for any finite group $G$ and any prime $p$.