Based on a sample of $1.31 \times 10^9$ $J/ψ$ events collected with the BESIII detector, an amplitude analysis of the isospin-violating decays $η^\prime → π^+π^-π^0$ and $η^\prime → π^0π^0π^0$ is performed. A significant $P$-wave contribution from $η^\prime → ρ^{±} π^{∓}$ is observed for the first time in $η^\prime → π^+π^-π^0$. The branching fraction is determined to be ${\mathcal B}(η^\prime → ρ^{±}π^{∓})=(7.44±0.60±1.26±1.84)\times 10^{-4}$, 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 ${\mathcal B}(η^\prime → π^+π^-π^0)_S=(37.63±0.77±2.22±4.48)\times 10^{-4}$ and ${\mathcal B}(η^\prime → π^0π^0π^0)=(35.22±0.82±2.54)\times 10^{-4}$, respectively. The latter one is consistent with previous BESIII measurements.