On the clique number of non-commuting graphs of certain groups

Let $G$ be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joint if and only if they do not commute. In a finite simple graph $\Gamma$ the maximum size of a complete subgraph of $\Gamma$ is called the clique number of $\Gamma$ and it is denoted by $\omega(\Gamma)$. In this paper we characterize all non-solvable groups $G$ with $\omega(\mathcal{A}_G)\leq 57$, where the number 57 is the clique number of the non-commuting graph of the projective special linear group $\mathrm{PSL}(2,7)$. We also complete the determination of $\omega(\mathcal{A}_G)$ for all finite minimal simple groups.