Eccoci.

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.