Custom field group weight order on contact summary page?
The 'order' only controls a field set in respect of other Custom Field sets, it does not relate to Core field sets that show on Summary. 

Eg on demo I just made a new field set for Contact. It showed to the right of Constituent Information.

I then edited the order to push my new set to the 'top' and it then showed on the left which Constituent Information then showed on the right.