My understanding of an ideal block cipher is as follows:
For a block cipher with block size $n$, there are $2^n$ possible plaintexts. The number of possible keys (mappings) would then be $2^n!$.
The size of a single key would then be $2^n$ because you're choosing one full permutation from the set of all possible permutations.
EDIT: Now that I think about it some more, the key size would be the required number of bits to express the number of possible permutations. So for 24 permutations, that would be a 5-bit key size. For 40,320 permutations, that would be a 16-bit key size.
Is this correct?