We demonstrate that the linear quotient singularity for the exceptional subgroup G
in Sp(4, C) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver.
This allows us to construct uniformly all 81 projective crepant resolutions of C4/G as hyperpolygon
spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa
Weyl group action via an explicit hyperplane arrangement. More generally, for the n-pointed star
shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon
spaces in dimension 2n − 6; for example, we show that there are 1684 projective crepant resolutions
when n = 6. We also prove that the resulting affine cones are not quotient singularities for n ≥ 6.