  Based on $(10087\pm44)\times10^{6}$ $J/\psi$ events collected with the BESIII
detector, the process $J/\psi\rightarrow\gamma\pi^{+}\pi^{-}\eta'$ is studied
using two dominant decay channels of the $\eta'$ meson,
$\eta'\rightarrow\gamma\pi^{+}\pi^{-}$ and
$\eta'\rightarrow\eta\pi^{+}\pi^{-},\eta\rightarrow\gamma\gamma$. The $X(2600)$
is observed with a statistical significance larger than 20$\sigma$ in the
$\pi^{+}\pi^{-}\eta'$ invariant mass spectrum, and it has a strong correlation
to a structure around 1.5$~\mathrm{GeV}$/{\it c}$^{2}$ in the $\pi^{+}\pi^{-}$
invariant mass spectrum. A simultaneous fit on the $\pi^{+}\pi^{-}\eta'$ and
$\pi^{+}\pi^{-}$ invariant mass spectra with the two $\eta'$ decay modes
indicates that the mass and width of the $X(2600)$ state are $2617.8\pm2.1
^{+18.2}_{-1.9}~\mathrm{MeV}$/{\it c}$^{2}$ and
$200\pm8^{+20}_{-17}~\mathrm{MeV}$, respectively. The corresponding branching
fractions are measured to be $B(J/\psi\rightarrow\gamma X(2600))\cdot
B(X(2600)\rightarrow f_{0}(1500)\eta')\cdot
B(f_{0}(1500)\rightarrow\pi^{+}\pi^{-})$ =
($3.39\pm0.18^{+0.91}_{-0.66})\times10^{-5}$ and $B(J/\psi\rightarrow\gamma
X(2600))\cdot B(X(2600)\rightarrow f_{2}^{'}(1525)\eta')\cdot
B(f_{2}^{'}(1525)\rightarrow\pi^{+}\pi^{-})$ =
($2.43\pm0.13^{+0.31}_{-1.11})\times10^{-5}$, where the first uncertainties are
statistical, and the second systematic.
