By a movement of a space with a geometrical structure we denote any transformation preserving this structure. Such transformations form a group which often is a Lie group. In this paper, the author proves that a linearly connected space of hyperplane elements of maximal mobility admits a movement group \(G_r\) possessing \(r=n^2+2\) parameters. The author establishes some necessary and sufficient conditions characterizing these spaces. The author gives exact bounds of the first lacuna \([n^2+1,n^2n-1]\) and shows that such a space with nonzero curvature may have a group \(G_r\) of maximal order with \(r=n^2\), \(n\geq 4\).