Let \(\mathrm E_6\) denote the simply-connected compact exceptional Lie group of rank 6. The Lie group \(\textrm{Spin}(10)\) naturally embeds in \(\mathrm E_6\) , corresponding to the inclusion of the Dynkin diagrams. We determine the K-ring of the coset space \(\mathrm E_6/\textrm{Spin}(10)\) . We identify the class of the tangent bundle of \(\mathrm E_6/\textrm{Spin}(10)\) in \(KO(\mathrm E_6/\textrm{Spin}(10))\) . As an application we show that \(\mathrm E_6/\textrm{Spin}(10)\) can be immersed in the Euclidean space \(\mathbb {R}^{53}\) but not in \(\mathbb {R}^{40}.\)
