Qtransitive positions are highly symmetric, and all Qtransitive positions are local maxima except for the Start position itself which might be described as a local minima. The idea of Qtransitive positions was introduced in a paper by Dan Hoey and Jim Saxbe. Their paper was written to answer the informal assertion that "all positions that are completely symmetric" are local maxima, where a local maximum is a position from which no move takes the Cube to a position that is further from Start. 
The problem with this informal assertion is that it leaves the question of what constitutes a completely symmetrical position ill defined. Really, it leaves the question of what constitutes a completely symmetrical position not defined at all. So the HoeySaxbe paper became the seminal work on defining symmetry in the context of Rubik's Cube. 
Symmclass(x) = M

In the quarter turn metric, any move takes a local maximum to a position that is one move closer to Start. In the face turn metric, a move may take a local maximum to a position that is one move closer to Start or to a position that is the same distance from Start. 
A strong local maximum in the face turn metric is a position from which all 18 face moves take the Cube one move closer to Start. A local maximum in the face turn metric that is not a strong local maximum is a weak local maximum. Most local maxima in the face turn metric seem to be weak local maxima. I'm not aware of any theoretical reason why strong local maxima must exist in the face turn metric, but a search of Cube space reveals that they do. Here follow the two shortest such positions. They were discovered by Jerry Bryan. 
D2 F2 L2 D' U L2 F2 D' U' (9f*)

U2 B2 L2 D U' R2 B2 D' U' (9f*)

This page last edited on 17 Jan 2013.