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Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let$\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over thefield $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalizedunits in $\mathbb{F}_pG$. Suppose that $G$ is a finite $p$-group given by acentral extension of the form $$1\longrightarrow \mathbb{Z}_{p^n}\times\mathbb{Z}_{p^m} \longrightarrow G \longrightarrow \mathbb{Z}_p\times\cdots\times \mathbb{Z}_p \longrightarrow 1$$ and $G'\cong \mathbb{Z}_p$, $n,m\geq 1$ and $p$ is odd. In this paper, the structure of $G$ is determined. Andthe relations of $V(\mathbb{F}_pG)^{p^l}$ and $G^{p^l}$,$\Omega_l(V(\mathbb{F}_pG))$ and $\Omega_l(G)$ are given. Furthermore, there isa direct proof for $V(\mathbb{F}_pG)^p\bigcap G=G^p$.

A class of finite $p$-groups and the normalized unit groups of group algebras