  The surface corresponding to the moduli space of quadratic endomorphisms of
$\mathbb{P}^1$ with a marked periodic point of order $n$ is studied. It is
shown that the surface is rational over $\mathbb{Q}$ when $n\le 5$ and is of
general type for $n=6$.
  An explicit description of the $n=6$ surface lets us find several infinite
families of quadratic endomorphisms $f: \mathbb{P}^1 \to \mathbb{P}^1$ defined
over $\mathbb{Q}$ with a rational periodic point of order $6$. In one of these
families, $f$ also has a rational fixed point, for a total of at least $7$
periodic and $7$ preperiodic points. This is in contrast with the polynomial
case, where it is conjectured that no polynomial endomorphism defined over
$\mathbb{Q}$ admits rational periodic points of order $n>3$.
