I'm working on a problem from Artin which asks to rule out as many class equations for a group of order 10 as I can. I'm are unsure about one:
1 + 1 + 2 + 2 + 2 + 2
My thought was that there are four conjugacy classes of order 2 each, so their corresponding centralizers will each have order five. Since they have prime order, we know the centralizers are all cyclic, so that implies they're the same. But what kind of group would have four conjugacy classes with the same centralizer?