We report a measurement of the branching fraction of $psi(3770)togammachi_{c1}$ and search for the transition $psi(3770)rightarrow gamma chi_{c2}$ based on 2.92~fb$^{-1}$ of $e^+e^-$ data accumulated at $sqrt{s}=3.773$~GeV with the BESIII detector at
the BEPCII collider. The measured branching fraction of $psi(3770) rightarrow gamma chi_{c1}$ is $mathcal B(psi(3770) rightarrow gamma chi_{c1})=(2.48 pm 0.15 pm 0.23) times 10^{-3}$, which is the most precise measurement to date. The upper limit on the branching fraction of $psi(3770)rightarrow gamma chi_{c2}$ at a $90%$ confidence level is $mathcal B(psi(3770) rightarrow gamma chi_{c2})<0.64 times 10^{-3}$. The corresponding partial widths are $Gamma(psi(3770) to gamma chi_{c1}) =(67.5pm 4.1pm 6.7)$~keV and $Gamma(psi(3770) to gamma chi_{c2}) < 17.4$~keV.
