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The group of automorphisms of S(5,8,24), M_{24}, is 5-transitive.

Other than Symmetric groups are there any other 5-transitive groups?

If not, would it be correct to say S(5,8,24) is the most symmetric object (not counting trivially obvious objects like the graph K_{n}) in existence?

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    $\begingroup$ The answer to the first question can easily be found on wikipedia: en.wikipedia.org/wiki/… (note that infinite groups are not considered here). The second question depends one how you define "most symmetric". $\endgroup$
    – Koen S
    Commented Nov 26, 2011 at 15:26
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    $\begingroup$ If you're looking for exceptional symmetries, you should broaden your horizons beyond graphs and codes. See for example the Leech lattice and the monster vertex algebra. $\endgroup$
    – S. Carnahan ♦
    Commented Nov 28, 2011 at 4:03

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If you're only interested in finite permutation groups, then Koen S has given you the answer you needed. If you allow infinite objects, then there are much more symmetric objects than S(5,8,24).

In fact, there is a notion of "highly transitive permutation groups": these are permutation groups (acting on an infinite set $\Omega$) that are $k$-transitive for every natural number $k$. Quite often, model-theoretic tools are used to construct non-obvious examples of such groups (e.g. using Fraïssé limits).

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