Hi there - I had a assessment centre at a strategy consultancy and have been working for them for about 2 and a half years ago and would say…
Group exercise: they are really seeing if you can work collaboratively in a team environment here, to see if you can respect others’ opinions but also make meaningful contributions yourself. My tip would be to have a few minutes where you lead the group for parts of the discussion - this will work to your advantage (leadership, taking the initiative), but make sure you do so in a collaborative / non-autocratic way and build consensus (think it will help if you reach some sort of conclusion at the end of the discussion you lead).
One on one case interview: from my experience, they wanted to see if I could go about solving a problem in a logical coherent way - I think CG would also be looking for grads who are able to approach the same problem from different angles, covering a wide range of potential issues that might arise from the case. Remember, the final ‘answer’ is of lesser importance here, it’s the approach that matters more. Prepare as much as possible (the material itself and lookup case studies on the web, there should be a lot of stuff!), structure your argument and be confident!
BEST OF LUCK (and let us know how it goes!!)