On the website GroupNames here that lists groups with orders below $500$ as well as the ATLAS of finite group representations (and other places I’m sure I’ve stumbled upon and forgotten about), conjugacy classes are often labelled as a number and then sometimes a letter. For example, here GroupNames labels $S_4$’s conjugacy classes as $1$, $2A$, $2B$, $3$, $4$. For a much longer example, here is ATLAS doing such a labelling on the baby monster.
What’s the meaning of the labels here? By the looks of it, it seems like the number corresponds to the largest algebraic degree of the entries of the character table for the column corresponding to that conjugacy class. Is that correct?
Furthermore, what’s the meaning of the letters? It seems the letters correspond to the size of the conjugacy class. For example, the baby monster link I provided shows it has conjugacy classes labelled $6A, \dots, 6K$ with $6A$ larger than $6B$, $6B$ larger than $6C$ and so on. Is this indeed the case? What if there were two conjugacy classes with the same number and same size - how are the letters chosen? (This can indeed happen, consider $A_5$, found here on GroupNames.)
And perhaps the most important question: (why) is such a labelling useful?
