Based on a sample of 1.31 x 10(9) J/Psi events collected with the BESIII detector, an amplitude analysis of the isospin-violating decays eta' -> pi(+)pi(-)pi(0) and eta' -> pi(0)pi(0)pi(0) is performed. A significant P-wave contribution from eta' -> rho(+/-)eta(-/+) is observed for the first time in eta' -> pi(+)pi(-)pi(0). The branching fraction is determined to be B(eta' -> rho(+/-)pi(-/+)) = (7.44 +/- 0.60 +/- 1.26 +/- 1.84) x 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 B(eta' -> pi(+)pi(-)pi(0))(S) = (37.63 +/- 0.77 +/- 2.22 +/- 4.48) x 10(-4) and B(eta' -> pi(0)pi(0)pi(0)) = (35.22 +/- 0.82 +/- 2.54) x 10(-4), respectively. The latter one is consistent with previous BESIII measurements.