  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
$\overline{\mathcal{M}}_{0,n}$.
