  A \emph{finite cover} of a group $G$ is a finite collection $\mathcal{C}$ of
proper subgroups of $G$ with the property that $\bigcup \mathcal{C} = G$. A
finite group admits a finite cover if and only if it is noncyclic. More
generally, it is known that a group admits a finite cover if and only if it has
a finite, noncyclic homomorphic image. If $\mathcal{C}$ is a finite cover of a
group $G$, and no cover of $G$ with fewer subgroups exists, then $\mathcal{C}$
is said to be a \emph{minimal cover} of $G$, and the cardinality of
$\mathcal{C}$ is called the \emph{covering number} of $G$, denoted by
$\sigma(G)$. Here we investigate the covering numbers of the McLaughlin
sporadic simple group and some low degree primitive groups.
