How about the Leech lattice. This is a 24-dimensional packing of unit spheres where each one touches 196560 others. It is the densest 24-dimensional lattice packing (and very likely the densest 24-dimensional sphere packing, although this has not been proved). It has a remarkable amount of symmetry, and most of the densest sphere packings known in dimensions < 24 are derived from it (and known sphere packings in dimensions > 24 are nowhere near as dense when normalized for the dimension).

Maybe this is already implicitly included in the list, as it is closely related to the monster vertex algebra.

How about the Leech lattice. This is a 24-dimensional packing of unit spheres where each one touches 196560 others. It is the densest 24-dimensional lattice packing (and very likely the densest 24-dimensional sphere packing, although this has not been proved - EDIT: it has been now). It has a remarkable amount of symmetry, and most of the densest sphere packings known in dimensions < 24 are derived from it (and known sphere packings in dimensions > 24 are nowhere near as dense when normalized for the dimension).

Maybe this is already implicitly included in the list, as it is closely related to the monster vertex algebra.