  For a finite group $G,$ $D(G)$ is defined as the least positive integer $k$
such that for every sequence $S=g_1 g_2\cdots g_k$ of length $k$ over $G$,
there exist $1 \le i_1 < i_2 <\cdots < i_m \le k $ such that $\prod_{j=1}^{m}
g_{i_{\sigma(j)}}=1$ holds for $\sigma = id,$ identity element of $S_m.$ For a
finite abelian group, this group invariant, known as the Davenport constant, is
crucial in the theory of non-unique factorization domains. The precise value of
this invariant, even for a finite abelian group of rank greater than $2$, is
not known yet. In 1977, Olson and White first worked with this invariant for
finite non-abelian groups. After that in 2004, Dimitrov dealt with it, where he
proved that $D(G)\leq L(G)$ for a finite $p$-group $G$, where $p$ is a prime
and $L(G)$ is the Loewy length of $\mathbb{F}_pG.$ He conjectured that equality
holds for all finite $p$-groups. In this article, we compute $D(G)$ for a
certain subclass of $2$-generated finite $p$-groups of nilpotency class two and
show that the conjecture is true by determining the precise value of the Loewy
length of $\mathbb{F}_pG.$ We also evaluate $D(G)$ for finite dicyclic,
semi-dihedral and some other groups.
