Need help determining what this center is isomorphic to I am looking at this released exam, problem 2c. It states:

Let $G$ be a non-abelian group of order $8$ and $Z$ be the center of $G$. To which group is $Z$ isomorphic?

It gives a hint to recall a theorem from what I believe is this released past midterm. I think he is referring to 4a, and 4b:

I'm still not exactly sure how to use them. As for b, $G$ is not of the form $pq$ it is $2^3$. And the first part, I'm not sure how I would fit that in.
I can "solve" it, because in the next question it is mentioned that $G/Z$ is isomorphic to Klein-4 (although I have not proven it). Hence $Z$ must have order $2$, as $G$ has order $8$ therefore $Z(G)$ divides $G$ and $8/2=4$. From this, I know there is only one possible group of order $2$, implying $Z \cong \mathbb{Z}_2$.
This is begging the question, as I don't know how to prove the one after this. And proving the original question would make the next question trivial.
Any hint in the right direction would be appreciated.

Hint: being a $p$-group, the center is nontrivial, and a group of order 2 is cyclic. What can the order of the center be?
