Using a low background data sample of 9.7×10^{5} J/ψ→γη^{'}, η^{'}→γπ^{+}π^{-} events, which are 2 orders of magnitude larger than those from the previous experiments, recorded with the BESIII detector at BEPCII, the decay dynamics of η^{'}→γπ^{+}π^{-} are studied with both model-dependent and model-independent approaches. The contributions of ω and the ρ(770)-ω interference are observed for the first time in the decays η^{'}→γπ^{+}π^{-} in both approaches. Additionally, a contribution from the box anomaly or the ρ(1450) resonance is required in the model-dependent approach, while the process specific part of the decay amplitude is determined in the model-independent approach.