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A partial-wave analysis is performed on the decay J/ψ → γπ0π0π0 within the π0π0π0 invariant-mass region below 1.6 GeV/c2, using (10.09 ± 0.04) × 109 J/ψ events collected with the BESIII detector. Significant isospin-violating decays of η(1405) and f1(1420) into f0(980)π0 are observed. For the first time, three axial vectors, f1(1285), f1(1420), and f1(1510), are observed to decay into π0π0π0. The product branching fractions of these resonances are reported.