  We study the radiative decay of $X(3872)$ based on the assumption that
$X(3872)$ is regarded as a $c\overline{c}$ charmonium with quantum number
$J^{PC}=1^{++}$ ($J,\,P,\,C$ represent the spin, parity and charge conjugation,
respectively). The form factors of $X(3872)$ transitions to $J/\psi\gamma$ and
$\psi'\gamma$ ($\psi'$ denotes $\psi(2S)$ throughout the paper) are calculated
in the framework of the covariant light-front quark model. The phenomenological
wave function of a meson depends on the parameter $\beta$, whose inverse
essentially describes the confinement scale. In the present work, the
parameters $\beta$ for the vector $J/\psi$ and $\psi'$ mesons will be
determined through their decay constants, which are obtained from the
experimental values of their partial decay widths to the electron-positron
pair. For $X(3872)$, we determined the value of $\beta$ by the decay width of
$X(3872)\rightarrow \psi'\gamma$. Then, we examined the width of $X(3872)\to
J/\psi\gamma$ in a manner of parameter-free prediction and compared it with the
experimental value. As a result, an inconsistency or contradiction occurs
between the widths of $X(3872)\to J/\psi\gamma$ and $X(3872)\to \psi'\gamma$.
We thus conclude that $X(3872)$ cannot be a pure $c\overline c$ resonance and
that other components are necessary in its wave function.
