A relation between $m_{G,N}$ and the Euler characteristic of the nerve space of some class poset of $G$

Let $G$ be a finite group and $N⊴G$ with $\mathrm{\mid }G:N\mathrm{\mid }=p$ for some prime $p$. In this note, to compute $m_{G,N}$ directly, we construct a class poset ${\mathfrak{T}}_C(G)$ of $G$ for some cyclic subgroup $C$. And we find a relation between $m_{G,N}$ and the Euler characteristic of the nerve space $\mathrm{\mid }N({\mathfrak{T}}_C(G))\mathrm{\mid }$ (see the Theorem 1.3). As an application, we compute $m_{S_5,A_5}=0$ directly, and get $S_5$ is a $B$-group.