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I don't see any explicit reference to this but I've noticed
that the simple group PSL(2,7) occurs naturally as a subgroup
of the 2x2x2 cube group of order 3674160 (well - with a slight
amount of wilful tinkering!). This is the model in which one
of the the 8 cubelets stays fixed.

One way of seeing how it is realised is to view to view the
corner cubelets as a single block, i.e. suppose all three
elements of each corner cubelet have the same colour.

Then taking the following labellings where all of 1 could be
coloured red, all of 2 yellow, etc. (UFR refers to the the
cubelet in the "Up" "Front" "Right" position, etc).