i wrote

B) What is the least commutative element (the anticentre?) of
the cube group?

i'll guess

corners:  (1)(7)     edges:    (1)(11)
corners:  (1+)(7-)   edges:    (1)(11)
corners:  (1-)(7+)   edges:    (1)(11)
corners:  (1)(7)     edges:    (1+)(11+)
corners:  (1+)(7-)   edges:    (1+)(11+)
corners:  (1-)(7+)   edges:    (1+)(11+)

after thinking about it, i realized that

corners:  (8)   edges:  (12)

commutes with even fewer elements. again, elements with
this cycle structure split into two conjugacy classes.

mike