In [Eur. J. Comb. 5, 59--91 (1984; Zbl 0574.20012)] \textit{M. Ronan} and the reviewer introduced a rank three minimal parabolic geometry for \(Co_2\). In this paper the authors investigate the point-line collinearity graph of this geometry. They gather a lot of results, among them the \(H\)-orbits of subgroups of \(Co_2\) isomorphic to \((2^{1+6}\times 2^4)A_8\), \(2^{1+8}Sp_6(2)\), \(2^{10}2^4S_6\), \(2^{1+8}U_4(2)2\), \(U_6(2)2\) and \(HS2\). This all will be done by embedding the graph into the Leech-lattice and then studying this lattice. The results then will be used in [J. Comb. Theory, Ser A 107, 181--213, 215--261 (2004; Zbl 1077.20029)], where the authors determine the point-line collinearity graph for the Baby Monster geometry.