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Just set this up on one of our servers so here is what i did

First, I went in to Exchange Management Console (2010 SP1) and created the Distribution Group.
Go to Recipient Configuration -> Distribution Group and select New Distribution Group from the Actions menu on the right side. Add the users that you want as recipients of that distribution list. For example, maybe it's the HR distribution group and so you'll add all members of your HR team. These are now the people that receive emails that are sent to this group.

Next, open Active Directory Users and Computers. Go to View and select Advanced Features. This will allow you to view the security options for your new Distribution Group.

Now, open the properties for your Distribution Group and you should see the security tab, open it. Click on Add and add the Users or Groups that you would like to have able to send as this group. The catch here is that you have to check the box for 'Send As' for each group or user that you just added. After this go to permissions for SELF and uncheck all boxes there.

You can then go to Advanced in the Security tab and select Effective Permissions to verify the Send As option for each user. Simply enter their name in and scroll down to Send As and see if it is checked.

Keep in mind, this may take up to a couple of hours to propagate through your domain, though for me it was instant.

Thanks for additional help from Spiceworks and mgmguy9958
https://community.spiceworks.com/how_to/2961-how-to-send-as-from-a-distribution-group-with-exchange-2010