Dear Rudolf,

a presentation that you can work with should have the subgroup U_6(2)
visible, then you can enumerate cosets of it in Co_2 without much
trouble. I don't know whether such an explicit presentation was ever
published, but it is implicit in characterisations of Co_2 via
a geometry called extended dual polar space, see e.g.
A. A. Ivanov, Exceptional Extended Dual Polar Spaces, Eur. J. Comb. 18
(1997),  859-885, or 
Hans Cuypers, Extended near hexagons and line systems, Advances in
Geometry 4 (2004), 181-214.

Hope this helps,
Dmitrii

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