In theory of finite simple groups, the following is a well celebrated
Theorem (Brauer-Fowler) If $G$ is a finite simple group and $z$ is an element of order $2$ such that centralizer of $z$ has order $n$ in $G$ then $G$ can be embedded in the alternating group $A_{n^2-1}$.
Question 1. Do any one lists an example(s) of simple group(s) in which $G$ can not be embedded in $A_{n^2-2}$?
Ignore next statements and question 2
Considering families of simple groups other than $A_n$, some simple groups in other families are isomorphic with some alternating groups (for example $\rm{PSL}_2(\mathbb{F}_5)\cong A_5$).
Question 2. Is there any example of a simple group $G$ in the family of sporadic groups or classical groups to which, by applying theorem of Brauer-Fowler, we get the embedding to be an isomorphism?
Both questions are almost similar, but I had put them in different form, concerning some special interests in the theorem stated.