I am fairly confident that the solution is:
"CHMDBELIJGKFA"

Here is my line of thought ...

To find the first letter, list all letters in order:

A[12!]
B[12!]
C[12!]
etc...

There are 12! permutations starting with A, and similarly with B, C, etc. 12!=479,001,600. From the beginning of A-block to the end of C-block there are 3*12!=1,437,004,800 perms. This is higher than the index we are looking for.

From the beginning of A-block to the end of B-block, there are 2*12!=958,003,200 perm(utation)s. Therefore, the first letter must start with C.

Now reformulate the problem. We are looking for the 1234567890 - 479,001,600 = 276,564,690th position of C-block. List the 2nd letters of C-block in order:

A[11!]
B[11!]
D[11!]
etc...

Note that C is excluded since we know it is the first letter.

So continue in this way by listing "blocks" of permutations, finding where the index should go, and choosing that letter as the next one.

Adjust the target index by subtracting all perms occuring before that block. Eventually the permutation is discovered by using this "sandwiching" search method.

-Yuchian