Minimal degree faithful permutation representations of some finite simple groups

Question

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 $Suz$.

Answer 1

It is more standard to write "isomorphic to" rather than "structurally identical with".

M: 97239461142009186000,

Fi24: 306936,

B: 13571955000,

Co1: 98280,

Suz: 1782.

The ATLAS of Finite Groups is a good source for such information, or its online version

If you looked at their Wikipedia pages you would have found this data for many of these groups.

Incidentally, the Baby Monster B was first proved to exist by Charles Sims, who constructed this representation of degree greater than $13 \times 10^9$ on a computer. There is a now an independent computer-free proof using the construction of M as an automorphism group of the Griess algebra.