  Higgs boson couplings with gauge bosons are probed through $e^-e^+
\rightarrow Z \gamma h$ in an effective Lagrangian framework. For this study
the beam polarization facility at the ILC along with the typical center-of-mass
energy of 500 GeV is considered. The reach of the ILC with an integrated
luminosity of 300 fb$^{-1}$ in the determination of $CP$-conserving parameters
is obtained. Sensitivity of the probe of each of these couplings in the
presence of other couplings is investigated. The most influential coupling
parameters are $\bar c_W=-\bar c_B$. Other parameters of significant effect are
$\bar c_{HW}$ and $\bar c_{HB}$. A detailed study of the various kinematic
distributions represents possibilities to disentangle the effect of some of
these couplings.
