Based on a sample of $1.31 \times 10^9$ $J/\psi$ events collected with the BESIII detector, an amplitude analysis of the isospin-violating decays $\eta^\prime \rightarrow \pi^+\pi^-\pi^0$ and $\eta^\prime \rightarrow \pi^0\pi^0\pi^0$ is performed. A significant $P$-wave contribution from $\eta^\prime \rightarrow \rho^{\pm} \pi^{\mp}$ is observed for the first time in $\eta^\prime \rightarrow \pi^+\pi^-\pi^0$. The branching fraction is determined to be ${\mathcal B}(\eta^\prime \rightarrow \rho^{\pm}\pi^{\mp})=(7.44\pm0.60\pm1.26\pm1.84)\times 10^{-4}$, 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 $\sigma$ meson component. The branching fractions of the combined $S$-wave components are determined to be ${\mathcal B}(\eta^\prime \rightarrow \pi^+\pi^-\pi^0)_S=(37.63\pm0.77\pm2.22\pm4.48)\times 10^{-4}$ and ${\mathcal B}(\eta^\prime \rightarrow \pi^0\pi^0\pi^0)=(35.22\pm0.82\pm2.54)\times 10^{-4}$, respectively. The latter one is consistent with previous BESIII measurements.