I want to know if the popular Sudoku puzzle is a Cayley table for a group.

Methods I've looked at: Someone I've spoken to told me they're not because counting the number of puzzle solutions against the number of tables with certain permutations of elements, rows and columns, the solutions are bigger than the tables, but I can't see why because I don't know how to count the different tables for a group of order 9, and then permute the rows, columns and elements in different ways. Also I believe the rotations/reflections will matter in comparing these numbers too. It would also be nice if there was a way to know if the operation is associative just from the table.