  Let $G$ be a finite subgroup of $\mbox{GL}(2)$ acting on
$\mathbf{A}^2\setminus\{0\}$ freely. The $G$-orbit Hilbert scheme
$G\mbox{-Hilb}(\mathbf{A}^2)$ is a minimal resolution of the quotient
$\mathbf{A}^2/G$. We determine the generator sheaf of the ideal defining the
universal $G$-cluster over $G\mbox{-Hilb}(\mathbf{A}^2)$, which somewhat
strengthens the well-known McKay correspondence for a finite subgroup of
$\mbox{SL}(2)$. We also study the quiver structure of
$G\mbox{-Hilb}(\mathbf{A}^2)$ at every $G$-cluster
$O_{Z_y}=O_{\mathbf{A}^2}/I_y$ in terms of a collection of sort of minimal
$G$-submodules of $O_{Z_y}$ (called mono-special $O_{\mathbf{A}^2}$-submodules)
and generating $G$-submodules of $I_y$.
