To answer this question often one of the following methods, which can easily be formalised , is used:

choose N points on the sphere	choose N points on the sphere
to construct a polyhedron (e.g. convex hull,tangential)	calculate the distance of each pair of points
evaluate it by its facets or edges.	evaluate these distancies

Here you can find background for searching. Also you find some applets to:

find good point-arrangements for:	construct point-sets on the sphere using:	view some results
well known problems	complete list of simple polyhedra (8 ≤ N ≤ 21)	applet
free defined problems	different construction methods	list from literature
point-to-point-distancies	icosahedral symmetry	obtained results