  Restrictions are placed on the presentation and holonomy algebra of a
K\"ahler group. Let $X$ be compact K\"ahler. It is known that the formality of
$X$, its Albanese map and the Hodge structure of $H^{\ast}(X)$ impose
restrictions on the holonomy algebra of $\pi_1 X$. We derive further
restrictions from Sullivan's 1-minimal model. An effective algorithm is given
to compute the holonomy algebra of a group from a presentation. Some very
strong restrictions on K\"ahler groups with few defining relations are
obtained.
