Based on (10087±44)×10^{6} J/ψ events collected with the BESIII detector, the process J/ψ→γπ^{+}π^{-}η^{'} is studied using two largest decay channels of the η^{'} meson, η^{'}→γπ^{+}π^{-} and η^{'}→ηπ^{+}π^{-}, η→γγ. A new resonance, which we denote as the X(2600), is observed with a statistical significance larger than 20σ in the π^{+}π^{-}η^{'} invariant mass spectrum, and it has a connection to a structure around 1.5 GeV/c^{2} in the π^{+}π^{-} invariant mass spectrum. A simultaneous fit on the π^{+}π^{-}η^{'} and π^{+}π^{-} invariant mass spectra with the two η^{'} decay modes indicates that the mass and width of the X(2600) state are 2618.3±2.0_{-1.4}^{+16.3} MeV/c^{2} and 195±5_{-17}^{+26} MeV, where the first uncertainties are statistical, and the second systematic.