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
abcdefghijklmnopqrstuvwxyz
qwertyuiopasdfghjklzxcvbnm
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
(aqjphioguxbwvcetzmdrk)(fyn)(ls)
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.
God plays dice 
Um, the g(26) in your link is 1260, double 630…
As David says, the actual largest order is 1260, achievable as lcm(4,5,7,9). Note that 4+5+7+9 = 25, so you don’t even use one of the letters.
But there’s a very natural permutation of shape 7-9-10: each letter goes to the one to the right of it on the keyboard, giving cycle notation (zxcvbnm)(asdfghjkl)(qwertyuiop).
I found a 10-7-9 type. Instead of taking QWERTY horizontally, go vertically.
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Q A Z W S X E D C R F V T G B Y H N U J M I K O L P
yielded cycles:
(AQHDWKFXOB)(CZPYLVI)(ESUMTJRNG)
The three rows of the keyboard have 10, 9, and 7 letters, respectively. Perhaps you thought of making the rows cycles? That is, taking the permutation (qwertyuiop)(asdfghjkl)(zxcvbnm).
The rows of a keyboard have 10, 9, and 7 letters, respectively. Were you thinking of (qwertyuiop)(asdfghjkl)(zxcvbnm), that is, the permutation whose cycles are the rows of a keyboard?
The 3 rows of letters on the keyboard have lengths 10, 9, and 7 – perhaps your construction used that fact?
I found a 10-7-9 type. Instead of taking QWERTY horizontally, go vertically.
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Q A Z W S X E D C R F V T G B Y H N U J M I K O L P
yielded cycles:
(AQHDWKFXOB)(CZPYLVI)(ESUMTJRNG)
If you replace the cycle of length 10 by cycles of length 5,4 and 1, you double the order of the permutation.
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