Another option is to add a table of Groups, with fields for FullName and ShortName, and get the short name from there via a relationship matching on full name. This way you avoid storing your data in a calculation formula.
If I read your logic correctly, you want to show the first letter of your 'Group' field, converted to uppercase?
If so, you don't need to use a 'Case' or 'If' function - just try the following formula:
Shortened Group Name = Upper(Left(Group;1))
(This picks the first letter of the contents of the field 'Group' and then converts it to uppercase.)
This will only work if, in every case, you require the initial letter of the 'Group' field in uppercase.
If you need a different initial letter in one or more cases, then you will need to use either of the other two suggestions that you have received so far.
Best wishes - Alan Stirling
This the definitely the best route if the Group only contains 1 item. If there are multiple group names, and you want, as the OP states, to check if group "contains/states" a value, use a case statement like this