Cubic permutations

The cube, \( 41063625 \) (\( 345^3 \)), can be permuted to produce two other cubes: \( 56623104 \) (\( 384^3 \)) and \( 66430125 \) (\( 405^3 \)). 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.