Quasirecognition by prime graph of $L_{10}(2)$

Let $G$ be a finite group. The prime graph of $G$ is denoted by $\Gamma(G)$. The main result we prove is as follows: If $G$ is a inite group such that $\Gamma(G)=\Gamma(L_{10}(2))$ then $G/O_2(G)$ is isomorphic to $L_{10}(2)$. In fact we obtain the first example of a finite group with the connected prime graph which is quasirecognizable by its prime graph. As a consequence of this result we can give a new proof for the fact that the simple group $L_{10}(2)$ is uniquely determined by the set of its element orders.