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Unfortunately the Generic Control renders the contents straight to the page's markup. It is not wrapped into a container div for example which can take the css class name. You have two options if you wish to do this:
1) when creating generic content go to html mode and wrap the whole content into a div with class name (not applicable to non technical users)
2) Create a control which inherits from generic content. This control will render the content of the control within a containing div. Then you can simply set this div to have a class name define rules for it. Sample on this can be found here: How to wrap a Generic Content Control
I would recommend the second option as you only need to apply this once.
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