It is a rather well known fact, that there exist universal finitely presented groups (finitely presented groups, that contain all other finitely presented groups as subgroups). It is a rather direct consequence of the following two theorems:
Every countably-generated recursively-presented group is a subgroup of some $2$-generated recursively-presented group. (Higman, Neuman, Neuman)
A finitely generated group is recursively presented iff it is a subgroup of some finitely-presented group. (Higman)
To prove the existence of universal finitely-presented group we need to recursively enumerate all finite presentations and then apply those two theorems consecutively to the free product of groups, described by them.
However, I would like to know, what is the "smallest possible" presentation that can describe such group.
Let's determine the question more strictly:
Suppose $\langle a_1, ... , a_n| w_1, ... , w_r\rangle$ is a group presentation. Let's define it's length as $n + \sum_{i=1}^r |w_r|$, where $|w|$ is the length of the word $w$.
What is the minimal possible length of a presentation defining a universal finitely-generated group?