I'm looking for examples of $p$-groups $G$ with the following three properties:

1. the center of $G$ is $\mathbb{Z}/p$, and

2. $G^{\text{ab}} = (\mathbb{Z}/p)^n$ for some $n$, and

3. for every $g \in G$ whose image in $G^{\text{ab}}$ is nonzero, the cyclic group generated by $g$ contains the center.

Easy examples include cyclic groups of order $p$ and the quaternion group.

I'm particularly interested in examples with large $n$.

I'm looking for examples of $p$-groups $G$ with the following three properties:

1. the center of $G$ is $\mathbb{Z}/p$, and

2. $G^{\text{ab}} = (\mathbb{Z}/p)^n$ for some $n$, and

3. for every $g \in G$ whose image in $G^{\text{ab}}$ is nonzero, the cyclic group generated by $g$ contains the center.

Easy examples include cyclic groups of order $p$ and the quaternion group.

I'm particularly interested in examples with large $n$.

EDIT: Stefan Kohl's answer is helpful, but not quite what I'm looking for. Let me ask a slightly more focused question: do there exist examples with $n$ arbitrarily large?