You can do better than that for a lower bound! Say you consider all
single-hand-motion twists to be okay. Then yes, there are 18 such,
but there's no point in twisting the same face twice consecutively, so
after the first twist the tree branching factor is only 15. In fact, there's
no point in twisting a face twice if the only intervening twist was done
on the opposite face; if we look at the operations of the form "twist face
X thusly and the opposite face thusly", there are 45 initial such operations,
and 30 at each succeeding branch, but since some branches now represent
two twists and some only one twist, it's much harder to compute the
minimum depth of the tree.

-- Don.