  Einstein's theory of general relativity has been extensively tested in weak
gravitational fields, mainly with experiments in the Solar System and
observations of radio pulsars, and current data agree well with the theoretical
predictions. Nevertheless, there are a number of scenarios beyond Einstein's
gravity that have the same predictions for weak fields and present deviations
only when gravity becomes strong. Here we try to test general relativity in the
strong field regime. We fit the X-ray spectrum of the supermassive black hole
in Ark 564 with a disk reflection model beyond Einstein's gravity, and we are
able to constrain the black hole spin $a_*$ and the Johannsen deformation
parameters $\alpha_{13}$ and $\alpha_{22}$ separately. For $\alpha_{22} = 0$,
we find $a_* > 0.96$ and $-1.0 < \alpha_{13} < 0.2$ with a 99% confidence
level. For $\alpha_{13} = 0$, we get $a_* > 0.96$ and $-0.1 < \alpha_{22} <
0.9$ with a 99% confidence level. Our measurements are thus consistent with the
hypothesis that the supermassive compact object in Ark 564 can be described by
the Kerr metric.
