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?



The Symmetry of Steiner System S(5,8,24)
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).


