1. Load the dimension table 5 times and associate each Code with the corresponding Dimension, or
2. Join the 5 Descriptions into the Fact table and add 5 Description fields to your Fact, or
3. Further normalize your fact table and create a seaprate relation table that connects the Fact rows to their Corresponding Code Groups (in your example, 5 rows per each Fact row).
Finding the best solution out of the threee would be a tough choice, driven by your business needs, the actual meaning of the 5 Groups, the size of your Fact table and the size of your Dimensional Tables.
For example, if your business requirement is to have a single field on the screen called "Group Description, then option 3 is probably your best choice. If, on the other hand, you need the 5 Group Descriptions in separate list boxes, then your choices are probably between 1 and two, depending on the data size.