Suppose $G$ is a finite $p$-group and $d$ is its minimal number of generators. If $G=\left< X | R \right>$, where $|X|=d$ and $|R|=r$, then $r \geq d^2/4$. So if $d>4$ you certainly cannot have an efficient presentation. (If you increase the number of generators, you will need also to increase the number of relations by at least the same amount).

I think this could somehow be applied to any finite group. However, you need to be more careful as you have to to talk about pro-$p$ presentations. So I have to think about it.

Edit: Thinking about it a bit more, I cannot see how to extend this to any finite group. Also, like Sam I think these are called balance presentations.

Suppose $G$ is a finite $p$-group and $d$ is its minimal number of generators. If $G=\left< X | R \right>$, where $|X|=d$ and $|R|=r$, then $r \geq d^2/4$. So if $d>4$ you certainly cannot have an efficient presentation. (If you increase the number of generators, you will need also to increase the number of relations by at least the same amount).

I think this could somehow be applied to any finite group. However, you need to be more careful as you have to to talk about pro-$p$ presentations. So I have to think about it.

Edit: Thinking about it a bit more, I cannot see how to extend this to any finite group. Also, like Sam I think these are called balance presentations.

Edit2: I forgot to mention that the inequality above is the Golod-Shafarevich inequality or at least a specifc case of it.

Suppose $G$ is a finite $p$-group and $d$ is its minimal number of generators. If $G=\left< X | R \right>$, where $|X|=d$ and $|R|=r$, then $r \geq d^2/4$. So if $d>4$ you certainly cannot have an efficient presentation. (If you increase the number of generators, you will need also to increase the number of relations by at least the same amount).

I think this could somehow be applied to any finite group. However, you need to be more careful as you have to to talk about pro-$p$ presentations. So I have to think about it.

Suppose $G$ is a finite $p$-group and $d$ is its minimal number of generators. If $G=\left< X | R \right>$, where $|X|=d$ and $|R|=r$, then $r \geq d^2/4$. So if $d>4$ you certainly cannot have an efficient presentation. (If you increase the number of generators, you will need also to increase the number of relations by at least the same amount).

I think this could somehow be applied to any finite group. However, you need to be more careful as you have to to talk about pro-$p$ presentations. So I have to think about it.

Edit: Thinking about it a bit more, I cannot see how to extend this to any finite group. Also, like Sam I think these are called balance presentations.