  The Hilbert Series (HS) of the moduli space of two G instantons on C^2, where
G is a simple gauge group, is studied in detail. For a given G, the moduli
space is a singular hyperKahler cone with a symmetry group U(2) \times G, where
U(2) is the natural symmetry group of C^2. Holomorphic functions on the moduli
space transform in irreducible representations of the symmetry group and hence
the Hilbert series admits a character expansion. For cases that G is a
classical group (of type A, B, C, or D), there is an ADHM construction which
allows us to compute the HS explicitly using a contour integral. For cases that
G is of E-type, recent index results allow for an explicit computation of the
HS. The character expansion can be expressed as an infinite sum which lives on
a Cartesian lattice that is generated by a small number of representations.
This structure persists for all G and allows for an explicit expressions of the
HS to all simple groups. For cases that G is of type G_2 or F_4, discrete
symmetries are enough to evaluate the HS exactly, even though neither ADHM
construction nor index is known for these cases.
