  The decays $\chi_{cJ}\to\gamma V$ ($V=\phi, \rho^0, \omega$) are studied with
a sample of radiative $\psip\to\gamma\chi_{cJ}$ events in a sample of
$(1.06\pm0.04)\times 10^{8} \psip$ events collected with the BESIII detector.
The branching fractions are determined to be: ${\cal B}(\chi_{c1}\to
\gamma\phi)=(25.8\pm 5.2\pm 2.3)\times 10^{-6}$, ${\cal B}(\chi_{c1}\to
\gamma\rho^0)=(228\pm 13\pm 22)\times 10^{-6}$, and ${\cal B}(\chi_{c1}\to
\gamma\omega)=(69.7\pm 7.2\pm 6.6)\times 10^{-6}$. The decay $\chi_{c1}\to
\gamma\phi$ is observed for the first time. Upper limits at the 90% confidence
level on the branching fractions for $\chi_{c0}$ and $\chict$ decays into these
final states are determined. In addition, the fractions of the transverse
polarization component of the vector meson in $\chi_{c1}\to \gamma V$ decays
are measured to be $0.29_{-0.12-0.09}^{+0.13+0.10}$ for $\chi_{c1}\to
\gamma\phi$, $0.158\pm 0.034^{+0.015}_{-0.014}$ for $\chi_{c1}\to
\gamma\rho^0$, and $0.247_{-0.087-0.026}^{+0.090+0.044}$ for $\chi_{c1}\to
\gamma\omega$, respectively. The first errors are statistical and the second
ones are systematic.
