How many keys does the Playfair Cipher have?

I was just studying the Playfair cipher and from what I've understood, it is just a slightly better version of a Caesar cipher, in that it isn't actually mono-alphabetic but rather the 'digrams' are mono-alphabetic. I believe that since it offers a larger combination i.e 26*26=676 digrams, it is better than a Caesar Cipher which would give us a maximum of twenty-six keys. So does this mean that the Playfair Cipher have 676 keys including the duplicate or is it larger than that ?

When we consider that a Playfair key consists of the alphabet (reduced to 25 letters) spread on a 5x5 square, that's $25!$ keys (another formulation consider any string to be a key; then strings leading to the same square are equivalent keys).
I conclude Playfair has $25!/{5^2}=620448401733239439360000\approx2^{79}$ distinct keys classes (a 79-bit-key cipher). At least another source agrees.
This is much less than an arbitrary permutation of digrams with distinct letters, which allows $(25\cdot24)!\approx2^{4678}$ keys.
Counting duplicate keys, it would be $25!$ (remember i=j in playfair). Basically the key is the 5x5 square. There are $25!$ permutations of the $25$ characters which populate that square.