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 of $\mathrm{E}_8$.

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 $\mathrm{E}_8$, which conjugacy class is it labeled by? In particular, what is the order of this class?

Later: It sounds like Hugh is proposing $$ \begin{array}{rrrrrrr} & & 4\\ 6 & 3 & 2 & 5 & 10 & 5 & 10 \end{array}$$

but the locations of the (6,3), and of (5,10,5,10) are just guesses.