How do I figure out the number of unique combinations
The number of possible unique tables for your application is $255!$.  The key term you will want to look up is circular permutation. 

The number of tables you can have without wrapping around is a permutation of the elements (also called a 'linear permutation').  The number of possible tables of a permutation of the elements is $n!$, which is why you (correctly) calculated the number of possible permutations of the table as $256!$.  

Because your use is circular, you would need to calculate the number of unique circular permutations, which is equal to $(n-1)!$.  So for the usage in your application where you have 256 elements, the number of possible tables is $(256-1)! = 255!$.