Literature on group theory of Rubik's Cube

You want to look at the book

David Joyner. Adventures with group theory: Rubik's cube, Merlin's machine, and other mathematical toys, 2nd ed., The Johns Hopkins Univer. Press, 2008.

The focus of the book is not exclusively the cube, but this is discussed in detail, and a few of the remaining open problems are listed as well. A generous bibliography is provided, pointing to some of the classical works that first studied the cube seriously from a group theoretic point of view, notably

D. Singmaster, Notes on Rubik’s magic cube, Enslow, 1981.

Joyner is an advocate for Open Source, and he has written programs in Sage to look at the cube in detail.

Besides this, you may want to look at the paper

Thomas Rokicki. Twenty-two moves suffice for Rubik’s Cube®. Math. Intelligencer, 32 (1), (2010), 33–40. (Behind a paywall, unfortunately).

This paper discusses fairly recent work trying to optimize the needed number of moves. Shortly after the paper was written, Roricki, Kociemba, Davidson, and Dethridge proved (with the aid of computers, and extending the approach of the paper) that $20$ moves is the largest number required (in the half-turn metric). This is discussed in their website, God's number is $20$.