A quick note about the homotopy groups of the Cayley plane: Mimura computed some of them. Specifically for *i*=8,9... he computed that $\pi_i(\mathbf{CaP}^2)$ equals **Z**, **Z**/2, **Z**/2, **Z**/24, 0, 0, **Z**/2, **Z**/120, (**Z**/2)$^{\oplus3}$, (**Z**/2)$^{\oplus4}$, **Z**/24$\oplus$ **Z**/2, **Z**/504$\oplus$ **Z**/2, 0, **Z**/6, **Z**/4, **Z**$\oplus$ **Z**/120$\oplus$ (**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