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Cubic permutations

Problem 62 Published on Friday, 30th January 2004, 06:00 pm; Solved by 26731;
Difficulty rating: 15%

The cube, 41063625 (3453), can be permuted to produce two other cubes: 56623104 (3843) and 66430125 (4053). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.

Find the smallest cube for which exactly five permutations of its digits are cube.