What are the 2125922464947725402112000 symmetries of a Rubik's Cube? In a recent talk, Marcus du Sautoy says there are 2125922464947725402112000 (2.1*10^24) symmetries of a Rubik's cube, but doesn't explicitly identify what qualifies as a symmetry.
What counts as a symmetry of the Rubik's cube?  Is it a thing like, "turn the top face once clockwise, then once counterclockwise"?  
How are these symmetries counted?
 A: Usually, something is called a "symmetry" if it leaves something invariant. For instance, the letter T has a symmetry in that you can take the mirror image with respect to its vertical axis and the shape remains the same.
In the case of the Rubik's cube, if you consider the coloured faces, there are no symmetries at all, since every move will change some of the faces. What is meant by "symmetries" in this case is operations that leave the "structure" of the cube invariant, i.e. that leave the cube in the same shape as before if you disregard the colours on the faces. These can be elementary operations, such as turning one face clockwise, or compound operations, such as the one you gave as an example, or more complex ones. The important point is that any sequences of operations that lead to the same end result are considered the same; for instance, if you turn a face clockwise three times, this is considered the same as turning it counter-clockwise once.
Since none of these operations leave the colours of the faces unchanged, and since they are considered different if and only if the end result is different, they can be counted by counting the number of different configurations into which they can bring the colours on the faces. So there's no need to think through all those gazillions of different sequences of moves; all you have to do is reason about which configurations of the faces are reachable through sequences of operations, and then count those.
Edit in reponse to the comment: Apologies for apparently giving an answer in the wrong "register"; it didn't seem from the phrasing of the question that you knew what a group was :-) Also apologies for not checking the numbers.
The number you cite is actually the total number of different positions of the cube pieces. Beyond the number of colour configurations reachable through turning faces, which is usually cited, this includes factors of $12$ for the number of different ways the pieces can be taken apart and put back together again, and $4^6$ for the number of different orientations of the central squares, which can't be distinguished from the colour markings. Of these $4^6=2^{12}$ different orientations of the central squares, $2^{11}$ can be reached without taking the cube apart. So denoting the number of configurations that's usually cited by $n$, you get


*

*$n$ configurations without taking the
cube apart and without marking the
central squares

*$2^{11}n$ configurations without taking apart but with marking

*$12n$ with taking apart but without marking and

*$2^{12}\cdot12n$ with taking apart and marking,


and the number you cited is that last number.
