  The resonant structure of the four pion final state in the decay $\tau \to
3\pi\pi^0\nu_\tau$ is analyzed using 4.27 million $\tau^+\tau^-$ pairs
collected by the CLEO II experiment. We search for second class currents in the
decay $\tau \to \omega\pi\nu_\tau$ using spin-parity analysis and establish an
upper limit on the non-vector current contribution. The mass and width of the
$\rho'$ resonance are extracted from a fit to the $\tau \to \omega\pi\nu_\tau$
spectral function. A partial wave analysis of the resonant structure of the
$\tau \to 3\pi\pi^0\nu_\tau$ decay is performed; the spectral decomposition of
the four pion system is dominated by the $\omega\pi$ and $a_1 \pi$ final
states.
