Pullbacks of $\kappa$ classes on $\overline{\mathcal{M}}_{0,n}$

The moduli space $\overline{\mathcal{M}}_{0,n}$ carries a codimension-$d$ cycle class $\kappa_{d}$. We consider the subspace $\mathcal{K}^{d}_{n}$ of $A^d(\overline{\mathcal{M}}_{0,n},\mathbb{Q})$ spanned by pullbacks of $\kappa_d$ via forgetful maps. We find a permutation basis for $\mathcal{K}^{d}_{n}$, and describe its annihilator under the intersection pairing in terms of $d$-dimensional boundary strata. As an application, we give a new permutation basis of the divisor class group of $\overline{\mathcal{M}}_{0,n}$.