  We define two variants $e(G)$, $f(G)$ of the Davenport constant $d(G)$ of a
finite group $G$, that is not necessarily abelian. These naturally arising
constants aid in computing $d(G)$ and are of potential independent interest. We
compute the constants $d(G)$, $e(G)$, $f(G)$ for some nonabelian groups G, and
demonstrate that, unlike abelian groups where these constants are identical,
they can each be distinct. As a byproduct of our results, we also obtain some
cases of a conjecture of J. Bass. We compute the $k$-th Davenport constant for
several classes of groups as well. We also make a conjecture on $f(G)$ for
metacyclic groups and provide evidence towards it.
