# Understanding a bound on a group presentation length

If $$G$$ is a group, by a presentation of a group $$G$$, I mean a representation of $$G$$ by generators and relations.

If we have that the bound on a presentation length is $$O({\left( {\log \;\left| G \right|} \right)^2})$$ [Babai et al.], and you have a presentation of length e.g. 136 and the order of your group is e.g. 20160, how do you know whether or not your length 136 is within the bound of $$O({\left( {\log \;20160} \right)^2})$$? My problem might be understanding the asymptotic notation, as I am not that familiar with it.

Saying that the length is $$O((\log |G|)^2)$$ means that there is an unknown constant $$K$$ such that the length is at most $$K(\log |G|)^2$$. Since we don't know what $$K$$ is, and the author of the author of the paper probably does not care, you cannot possibly judge the correctness of the assertion from a single group $$G$$.
The information from a group $$G$$ only enables you to find a lower bound for the constant, so you would need a range of examples of different sizes $$|G|$$ in order to make a sensible assessment of $$K$$ and whether the bound is correct.