Can every finite group be presented as an Fp group?

Question

If we have a finite group $G$, then we know that $G$ is a quotient of some free group, say $F$, on some free set, say $X$.

Now, let $N$ have this: $G\cong F/N$, where $N$ is the normal closure of $R$ (some finite set of relators) in $F$.

Is it "safe" to say that $G \cong \langle X |R \rangle $ (or even $G = \langle X |R \rangle $)? Which leads to say $G$ is finitely presented group!

Thanks for advance.

Answer 1

Let $G$ be a finite group, and so its Cayley table is finite. Construct finitely presented group $F$ with generators elements in $G$, and relators $abc^{-1}$ if ab=c in the Cayley table.