You need execute permission on every directory in the path leading to the destination directory. While I think the read bit is not necessary on these intermediate directories, I suggest having that permission also, unless you have a specific reason not to.
Also, be aware than when a user is added to a group, that doesn't take effect for the current logins. I.e. if you add yourself to a group, you have to log out and log back in before that takes effect. If you want to see what groups your current login can access, type the groups command.