A finitely presented group which has more generators than relations has an infinite abelianization and so is an infinite group. Therefore, for a finite group, all presentations must have at least as many relations as generators. A presentation for a finite group is called efficient if the number of generators equals the number of relations.

I know the following examples of efficient presentations for finite groups with $2$ generators:

$< x,y | x^{2} = y^{2} = (xy)^{n}>$, for any $n \geq 2$, which is a group of order $4n^{2}-4n$ in which $xy$ has order $2n^{2}-2n$ and conjugating through $x$ or $y$ sends $xy$ to $(xy)^{2n-1}$. (When $n=2$, this is the quaternion group of order $8$.)

$< x,y | x^{3} = y^{3} = (xy)^{2}>$, which is a double cover of $A_{4}$ isomorphic to $SL(2,3)$. This group is isomorphic to the group of quaternion units, together with $\pm \frac{1}{2} \pm \frac{i}{2} \pm \frac{j}{2} \pm \frac{k}{2}$.

$< x,y | x^{4} = y^{3} = (xy)^{2}>$, which is a double cover of $S_{4}$ not isomorphic to $GL(2,3)$. This is isomorphic to the preceding group, together with every quaternion of the form $\pm \frac{u}{\sqrt{2}} \pm \frac{v}{\sqrt{2}}$, where $u$ and $v$ are two distinct elements of the set $\{ 1, i, j, k \}$.

$< x,y | x^{5} = y^{3} = (xy)^{2}>$, which is a double cover of $A_{5}$ isomorphic to $SL(2,5)$. This is isomorphic to the group of unit quaternions representing $SL(2,3)$ as above, together with a set of icosians explicitly described in Conway and Sloane's SPLAG, which I am too lazy to describe here.

All of these groups have a central element of order $2$, and except for the members of the infinite family that have $n$ being odd, a subgroup isomorphic to the quaternion group of order $8$.

How does this generalize to larger numbers of generators, or, at least, what are some examples of efficient presentations of finite groups using $3$ generators? (I need not explain why I know all examples of efficiently presented groups with a single generator.) Is there a group analogous to the quaternion group of order $8$ for each number of generators? Is there a group analogous to $SU(2)$, which has subgroups isomorphic to the smallest member of the infinite family and all three exceptional examples? Is the center of a nontrivial finite group possessing an efficient presentation always nontrivial? (And what is a good reference for discussing this?)