Homotopy groups of the stunted projective space through degree nine (${\pi }_7({\mathbb{RP}}^{10}/{\mathbb{RP}}^6)={\pi }_9({\mathbb{RP}}^{10}/{\mathbb{RP}}^6)=\mathbb{Z}/2$)

For this quotient, homotopy groups through degree six and in degree eight vanish. The degree-seven and degree-nine groups are $\mathbb{Z}/2$. A map to $K(\mathbb{Z}/2,7)$ induces an isomorphism on degree-nine integral homology because the relevant square of the bottom mod-two class is nonzero. Two relative Hurewicz steps then compute degrees eight and nine.