  The Andrews-Curtis conjecture remains one of the outstanding open problems in
combinatorial group theory. It claims that every normally generating $r$-tuple
of a free group $F_r$ of rank $r\geq 2$ can be reduced to a basis by means of
Nielsen transformations and arbitrary conjugations. These transformations
generate the so-called Andrews-Curtis group AC($F_r$). The groups AC($F_r$) ($r
= 2, 3, \ldots$) are actively investigated and allows various generalizations,
for which there are a number of results. At the same time, almost nothing is
known about the structure and properties of the original groups AC($F_r$). In
this paper we define a class $\{A_{r, s}: r, s \geq 1\}$ of generalized
Andrews-Curtis groups in which any group $A_{r,r}$ is isomorphic to the
Andrews-Curtis group AC($F_r$). We prove that every group $A_{2,s}$\, ($s \geq
1$) is non-finitely presented. Hence the Andrews-Curtis group AC($F_2$) $\simeq
A_{2,2}$ is non-finitely presented. Thus, we give a partial answer to the
well-known question about the finite presentability of the groups AC($F_r$),
explicitly stated by J. Swan and A. Lisitsa in the Kourovka notebook \cite{KN}
(Question 18.89).

Group Theory