  We present the first determination of the hadronic decays of the lightest
exotic $J^{PC}=1^{-+}$ resonance in lattice QCD. Working with SU(3) flavor
symmetry, where the up, down and strange quark masses approximately match the
physical strange-quark mass giving $m_\pi \sim 700$ MeV, we compute
finite-volume spectra on six lattice volumes which constrain a scattering
system featuring eight coupled channels. Analytically continuing the scattering
amplitudes into the complex energy plane, we find a pole singularity
corresponding to a narrow resonance which shows relatively weak coupling to the
open pseudoscalar--pseudoscalar, vector--pseudoscalar and vector--vector decay
channels, but large couplings to at least one kinematically-closed
axial-vector--pseudoscalar channel. Attempting a simple extrapolation of the
couplings to physical light-quark mass suggests a broad $\pi_1$ resonance
decaying dominantly through the $b_1 \pi$ mode with much smaller decays into
$f_1 \pi$, $\rho \pi$, $\eta' \pi$ and $\eta \pi$. A large total width is
potentially in agreement with the experimental $\pi_1(1564)$ candidate state,
observed in $\eta \pi$, $\eta' \pi$, which we suggest may be heavily suppressed
decay channels.
