  Using a data sample of $(1.0087\pm 0.0044)\times 10^{10}$ $J/\psi$ events
collected with the BESIII detector, the decays of $J/\psi\to\gamma\pi^{0}
(\eta, \eta^\prime)\to\gamma\gamma\gamma$ are studied. Newly measured branching
fractions are $\mathcal{B}$$(J/\psi\to\gamma\pi^{0})$=$(3.34\pm 0.02\pm
0.09)\times 10^{-5}$, $\mathcal{B}$$(J/\psi\to\gamma\eta)$=$(1.096\pm
0.001\pm0.019)\times 10^{-3}$ and
$\mathcal{B}$$(J/\psi\to\gamma\eta^\prime)$=$(5.40\pm 0.01\pm0.11)\times
10^{-3}$, where the first uncertainties are statistical and the second are
systematic. These results are consistent with the world average values within
two standard deviations. The ratio of partial widths
$\Gamma(J/\psi\to\gamma\eta^\prime)/\Gamma(J/\psi\to\gamma\eta)$ is measured to
be $4.93 \pm 0.13$. The singlet-octet pseudoscalar mixing angle $\theta_P$ is
determined to be $\theta_P = -(22.11 \pm0.26)^\circ$ or $-(19.34 \pm
0.34)^\circ$ with two different phenomenological models.
