The symmetric group $S_n$ - the group of permutations of an $n$-set - plays a very important role in Group Theory. The paramount importance of this group resides in the following fact: given any finite group $G$, there is a value of n such that $S_n$ possesses a subgroup that is structurally identical with $G$. The above statement is a fact.
Question: What is the minimum value of n such that $S_n$ possesses a subgroup that is structurally identical with the monster group $M$ (respectively other groups $Fi24,B,Co1, Suz$)?
Edit: From the comments. The question is as follows:
Question: Are the minimal degree permutation representations of the sporadic simple groups all known? I am particularly interested in the cases $M, Fi24, B, Co1$ and $Suz$.