  The NHEG algebra is an extension of Virasoro introduced in
[arXiv:1503.07861]; it describes the symplectic symmetries of
$(n+4)$-dimensional Near Horizon Extremal Geometries with $SL(2,R)\times
U(1)^{n+1}$ isometry. In this work we construct the NHEG group and classify the
(coadjoint) orbits of its action on phase space. As we show, the group consists
of maps from an $n$-torus to the Virasoro group, so its orbits are bundles of
standard Virasoro coadjoint orbits over $T^n$. We also describe the unitary
representations that are expected to follow from the quantization of these
orbits, and display their characters. Along the way we show that the NHEG
algebra can be built from u(1) currents using a twisted Sugawara construction.
