Minimal number of relations in finite 2-groups with 2, 3, and 4 generators

Question

I have received helpful answers to my two previous questions that focused on the symmetric group of degree 3 and the dihedral group of order 8. If $d$ is the minimal number of generators of a finite group, and $r$ is the minimal number of relations of the group, then in both examples $d = 2$, but for the given symmetric group we have $r = 2$ and for the given dihedral group we have $r = 3$. This holds for some generalizations of these groups as well.

If we focus now on 2-groups, I would like to know if there are any examples of finite 2-groups with $d = 2$ and $r = 2$, $d = 3$ and $r = 3$, or $d = 4$ and $r = 5$. This all goes back to sharpening the Golod/Shafarevich inequality, which says that for finite $p$-groups, $r > \large\frac{d^2}{4}$. I would like to know if in some particular cases (in particular if $d = 4$) if we can sharpen this further as $r > {\large\frac{d^2}{4}} + 1$ when $p = 2$.

Answer 1

Since asking my question I have learned that an example of a finite 2-group with d = 2 and r = 2 is the Quaternion group of order 8, with the presentation {a, b: a squared = b squared, (a inverse)ba = b inverse. I have also learned that it is known that there are finite 2-groups with d = 3 and r = 3 of orders 512, 1024 and 2048 (resp. 2 to the 9th, 10th, and 11th powers), as described in papers by authors including Fouladi, Jamali, Orfi, Havas, Newman, and O'Brien in 2004, 2008 and 2012 papers. Finally I have learned that a finite 2-group with d = 4 and r = 5 of order 16382 (2 the 14th power) was exhibited in a 2004 paper by Havas, Newman, and O'Brian, which was preceded by Havas and Newman in 1983 demonstrating four groups of the above type with orders 2 to the 16th, 17th, 18th, and 19th powers. Thus I have now answered all the questions I initially asked, although I don't like the answer that there exists a finite 2-group with d = 4 and r = 5, because it means that one cannot solve the infinite 2-class field tower conjecture for imaginary quadratic number fields with 2-class group of rank 4 by trying to sharpen the sharpened Golod/Shafarevich inequality. Oh well--such is math!