I’m reasonably sure that long ago and totally by accident, I discovered a permutation of the alphabet a, b, …, z that somehow naturally arose from the order of letters on the QWERTY keyboard and had order 630. One such permutation would be (abcdefg)(hijklmnop)(qrstuvwxyz), which has cycles of order 7, 9, and 10 and therefore has order the least common multiple of 7, 9, and 10, which is 630. But of course this doesn’t naturally arise from the keyboard. 630 is interesting here because it’s
the largest order of a permutation of 26 elements fairly large for the order of a permutation of 26 elements; the maximum is twice this, 1260, as pointed out by several commenters.
I had thought that this permutation was the one that, in the two-line notation, is written
which takes a to q, b to w, and so on. But I checked during an idle moment earlier today; rewriting this in the cycle notation gives
which has cycles of length 21, 3, and 2 and therefore has order lcm(21, 3, 2) = 42. So what was I thinking of?
Answer, added Wednesday, May 2: instead of going horizontally, go vertically: the second line is qazwsxedcrfvtgbyhnujmikolp, which gives the 7-9-10 cycle type.