Let $G \subset \mathrm{SL}_2(\mathbf{C}^2)$ be a finite subgroup isomorphic to the binary icosahedral group. Let $Y$ be the minimal resolution of $\mathbf{C}^2/G$. The irreducible components of the exceptional fiber of $Y$ are naturally in correspondence with nodes of the Dynkin diagram.

Each of these components has a linking circle in $(\mathbf{C}^2 - \{0\})/G$, whose fundamental group is $G$. Thus, each component determines a nontrivial conjugacy class in $G$.

There are 8 nontrivial conjugacy classes in $G$, of orders 2,3,4,5,5,6,10,10. For each node of $E_8$, which conjugacy class is it labeled by? In particular, what is the order of this class?