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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   $n&gt;3$  .Communicated by Umberto Zannier

Moduli Spaces of Quadratic Rational Maps with a Marked Periodic Point of Small Order