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