The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is nonnegative for all closed simply connected manifolds of dimension $\ge 5$. We extend this to show that Yamabe invariant is nonnegative for all closed manifolds of dimension $\ge 5$ with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed oriented manifolds of dimension $\ge 5$ with non-spin universal cover, with finite fundamental group having all Sylow subgroups elementary abelian, admit metrics of \psc, once one restricts to the ``complement'' of manifolds whose homology classes are ``toral.'' The exceptional toral homology classes only exist in dimensions not exceeding the ``rank'' of the fundamental group, so this proves important cases of the Gromov-Lawson-Rosenberg Conjecture once the dimension is sufficiently large.