Number whose first six multiples permute its digits

Find a positive integer that can be multiplied by each of $1,2,3,4,5$ and $6$ without any new digit appearing in the products (each product uses only digits of the original number).

$142857$. It is the repeating block of $\tfrac17=0.\overline{142857}$, and $k\cdot142857$ for $k=1,\dots,6$ is the repeating block of $\tfrac{k}{7}$, which is a cyclic shift of the same six digits (e.g. $2\times142857=285714$).