I watched the video, and I believe I understand it now. The number he is asking about is the number of symmetries of Rubik's cube. This is not the number of symmetrical positions.
He mentions the number started with a 2, followed by 24 digits. So I'm pretty sure the number Robert mentioned is the right number. But I believe it is arrived at using |G|*2048*24. Here |G| means the size of the standard Rubik's cube group (<U,D,L,R,F,B>) which we know is 43252003274489856000. It's multiplied by 2048 because that's the number of it's possible for the 6 centers to be rotated that many combination of ways for a given element of G. In other words, |G|*2048 is the number of positions for the 3x3x3 supercube. Finally, it is multiplied by 24 because that is the number of ways the Rubik's cube can be reoriented.
Now to explain more why I think that is the right answer.
In the lecture, he talks about the symmetries of two objects with six symmetries, an equilateral triangle, and a starfish (one with 6 arms apparently). The triangle can be rotated 3 ways in its plane to leave it looking the same, or it can be reflected (or rotated so you view it from its other side) or not reflected for another factor of 2 symmetries. The starfish being a phyiscal object can not be transformed into a reflected starfish, and if you rotate it to view it from its other side, it doesn't look the same. Similarly, the (geometrical) cube has 48 symmetries. We can reorient it 24 ways without changing its appearance, and we can reflect it without changing its appearance. (Because of its 3-dimensional nature, reflecting it is not the same as viewing it from the other side.)
So now the Rubik's cube. Basically, he asked something like "How many ways can we pick it up, manipulate it, and set it back down so that it looks the same as it did before it was picked up?" (Well, we have to consider the cube with its colored stickers removed to have it really look the same.) Without "cheating" (disassembling and reassembling the cube, for instance), we can move the corners around edges around |G| ways with respect to the centers. If we know how the corners and edges have been permuted, we know that there are 2048 combinations of ways the centers can be rotated. Since these are physically realizable changes we can make to the cube (and don't change the appearance of the cube), these count as symmetries. Finally, when we set the cube back down there are 24 different orientations we can place it down without affecting how the cube looks. (We can't, for instance, put it down rotated horizontally 45 degrees, because then it will appear to have moved from how we had it originally.) Like a starfish, we can't physically reflect it. We can rotate it to view it from the other side, but we've already counted that. So |G|*2048*24 is the number of ways we can manipulate it without changing how it appears.
