Berger showed the homogeneous space $SU(5)/Sp(2)\cdot S^1$ has a metric of positive sectional curvature. Here, $Sp(2)\cdot S^1 = Sp(2)\times_{\mathbb{Z}/2\mathbb{Z}} S^1$. More generally, Bazaikin found an infinite family of free isometric actions of $Sp(2)\cdot S^1$ on $SU(5)$ (with appropriate left invariant metric), for which infinitely many of the quotients inherit a metric of positive sectional curvature.
The fact that the group acting is $Sp(2)\cdot S^1$ and not $Sp(2)\times S^1$ caused a few headaches when it came time to compute the topology of these examples.