Link between centrosymmetry, dipole moment and chirality
A molecule can't be chiral if it has an $S_n$ improper rotation (note that $S_1$ is equivalent to a mirror plane and $S_2$ is equivalent to inversion). Chirality is only possible in the $C_n$ and $D_n$, as well the less common $T$, $O$, and $I$, point groups, which only have rotation axes as symmetry elements.

A dipole moment can only occur along the primary $C_n$ rotation axis of a molecule (if it has one) and if there are any perpendicular mirror planes or rotation axes, it will have to be zero by symmetry. A $S_n$ improper rotation with $n&gt;1$ also precludes a dipole. This limits the point groups that can have dipoles to $C_n$, $C_{nv}$, and $C_s$.

Centrosymmetry is equivalent to having the $S_2$ symmetry operation. As mentioned above, this eliminates the possibility of chirality or a dipole. However, centrosymmetry is just a special case and  molecule won't be chiral and/or won't have a dipole under other conditions as well. That is to say, centrosymmetry is sufficient, but not necessary for eliminating chirality and/or a dipole.

To summarize:
$$\begin{array}{|c|c|c|} \hline
\text{Condition} &amp; \text{Exact Point Group} \\ \hline
\text{Chiral/Nonpolar} &amp; I,O,T,D_{n} \\
\text{Achiral/Polar} &amp; C_{nv}, C_s  \\
\text{Chiral/Polar} &amp; C_n \\
\hline
\end{array}$$

Where I'm taking "exact point group" to mean the highest symmetry group the molecule is contained in. This delineation has some minor exceptions. Technically, molecules of the appropriate symmetry still might be nonpolar (or at least too small to measure) by an accidental cancellation. For chirality, while all molecules of these symmetries are formally chiral, they can appear achiral in practice if the enantiomers can readily interconvert, as occurs with nitrogen inversion.