How can I fully understand what are conjugacy classes are in groups?
I know the definition, that $a$ and $b$ are conjugate if $gag^{-1}=b$ for some $g\in G$. But what is the intuition? Using a multiplication table (all the possible multiplications), how can I understand what a conjugacy class is?
For example, on Wikipedia you can see this table:
How can I relate the idea of a conjugacy class with this table?