Based on a sample of 1.31×10^{9} J/ψ events collected with the BESIII detector, an amplitude analysis of the isospin-violating decays η^{'}→π^{+}π^{-}π^{0} and η^{'}→π^{0}π^{0}π^{0} is performed. A significant P-wave contribution from η^{'}→ρ^{±}π^{∓} is observed for the first time in η^{'}→π^{+}π^{-}π^{0}. The branching fraction is determined to be B(η^{'}→ρ^{±}π^{∓})=(7.44±0.60±1.26±1.84)×10^{-4}, where the first uncertainty is statistical, the second systematic, and the third model dependent. In addition to the nonresonant S-wave component, there is a significant σ meson component. The branching fractions of the combined S-wave components are determined to be B(η^{'}→π^{+}π^{-}π^{0})_{S}=(37.63±0.77±2.22±4.48)×10^{-4} and B(η^{'}→π^{0}π^{0}π^{0})=(35.22±0.82±2.54)×10^{-4}, respectively. The latter one is consistent with previous BESIII measurements.