Counting conjugacy classes of elements of finite order in Lie groups

Using combinatorial techniques, we answer two questions about simple classical Lie groups. Define N(G,m)N(G,m) to be the number of conjugacy classes of elements of finite order mm in a Lie group GG, and N(G,m,s)N(G,m,s) to be the number of such classes whose elements have ss distinct eigenvalues or conjugate pairs of eigenvalues. What is N(G,m)N(G,m) for GG a unitary, orthogonal, or symplectic group? What is N(G,m,s)N(G,m,s) for these groups? For some cases, the first question was answered a few decades ago via group-theoretic techniques. It appears that the second question has not been asked before; here it is inspired by questions related to enumeration of vacua in string theory. Our combinatorial methods allow us to answer both questions.