A characterization of the simple sporadic groups

Let $G$ be a finite group, $n_{p}(G)$ be the number of Sylow $p$–subgroup of $G$ and $t(2, G)$ be the maximal number of vertices in cocliques of the prime graph of $G$ containing 2. In this paper we prove that if $G$ is a centerless group with $t(2,G)\geq 2$ and $n_{p}(G)$=$n_{p}(S)$ for every prime $p\in \pi (G)$, where $S$ is the sporadic simple groups, then $S\leq G\leq$Aut$(S)$.