In base 10, the recurring bits of the fractions $\frac{1}7,\ldots,\frac{6}7$ are cyclic permutations of each other. e.g. $\frac{1}{7}=0.(142857)$
$\frac{2}{7}=0.(285714)$
$\frac{3}{7}=0.(428571)$
In which bases does there exist $n$ such that the recurring bits of the fractions $\frac{1}{n},\ldots,\frac{n-1}{n}$ are cyclic permutations of each other?