Paw Swap

Nine discs (or disks) with the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 are arranged in increasing order on a table. You can only swap the positions of 2 discs. For example, you may swap 1 with 5 to form 523,416,789. This is considered 1 move. You are invited to swap the discs to form a square number. What is the square root of the square number formed when you swap the discs the least number of times?

For example, you swap 7 with 3, 3 with 4, 4 with 8, and 2 with 5 to form 157,326,849, which is the square of 12, 543. But it can be done in fewer moves.

Sources: The Puzzle Mountain (Gyles Brandreth) Level 90- For the Numerate, Page 174 Question 9.