  Let $G$ be a finite group and $N\unlhd G$ with $|G: N|=p$ 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
$|N(\mathfrak{T}_{C}(G))|$ (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.
