A quick note about the homotopy groups of the Cayley plane: Mimura computed some of them. Specifically for $i=8,9\dots$ he computed that $\pi_i(\mathbf{CaP}^2)$ equals $\mathbf{Z}$, $\mathbf{Z}/2$, $\mathbf{Z}/2$, $\mathbf{Z}/24$, $0$, $0$, $\mathbf{Z}/2$, $\mathbf{Z}/120$, $(\mathbf{Z}/2)^{\oplus3}$,$(\mathbf{Z}/2)^{\oplus4}$, $\mathbf{Z}/24\oplus \mathbf{Z}/2$, $\mathbf{Z}/504 \oplus \mathbf{Z}/2$, $0$, $\mathbf{Z}/6$, $\mathbf{Z}/4$, $\mathbf{Z} \oplus \mathbf{Z}/120 \oplus (\mathbf{Z}/2)^{\oplus2}$, respectively. See Theorem 7.2 of his 1967 paper The homotopy groups of Lie groups of low rank:
http://www.ams.org/mathscinet-getitem?mr=206958
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.kjm/1250524375