  We obtain certain results on a finite $p$-group whose central automorphisms
are all class preserving. In particular, we prove that if $G$ is a finite
$p$-group whose central automorphisms are all class preserving, then $d(G)$ is
even, where $d(G)$ denotes the number of elements in any minimal generating set
for $G$. As an application of these results, we obtain some results regarding
finite $p$-groups whose automorphisms are all class preserving. In particular,
we prove that if $G$ is a finite $p$-groups whose automorphisms are all class
preserving, then order of $G$ is at least $p^8$ and the order of the
automorphism group of $G$ is at least $p^12$.

Group Theory