Customer dims get big, you need something to roll-up customers before showing as an attribute, no-one likes 20,000 columns in a dashboard or report.
I look for things to group by, like Geography, maybe Country, State, or Region, or demographic data like Male/Female, Age-bucket, etc.
Sounds like you have a good natural candidate with Group, I would add GroupKey to your CustomerDim, and when you present your CustomerDim, expose GroupName as a top-level attribute (hierarchy), then allow drill-through to State maybe, and lastly individual Customers.
Hope that helps,
ChrisBecker.blog