The vertex coordinate set for the contact polytope of the Leech lattice listed on Wikipedia contains all permutations of:

• $\{4,-4,0^{22}\}$
• $\{-3,1^{23}\}$
• $\{3,-1^{23}\}$

The convex hull of these 600 vertices is an expanded 24-simplex, meaning that there is a compound of expanded 24-simplices using the vertices and long edges of the Leech lattice’s contact polytope. How many expanded simplices are in this compound?