A Very Special Number

Computer Science Level 4

Find the smallest positive integer $n$ such that the first 10 digits of $\sqrt[3]{n}$ contain all the digits from 0 to 9 exactly once each.

Clarification: For example, the first 4 digits of $\sqrt[3]{2} = 1.25992\ldots$ are 1, 2, 5, and 9.