Hi Mercyl,
I just wrote a really long reply and lost it when I went to get this link for your

The link is a great number of pdf documents from a Government Professional Development package for teachers on Gifted Ed. it is FANTASTIC. Very lengthy but very well worth the read.
http://www.dest.gov.au/sectors/school_education/publications_resources/profiles/Gifted_Education_Professional_Development_Package.htm
I think you are on the right track and speaking to your daughter's teacher is the best place to start. Testing is definitely going to be a benefit. We had our son tested and it has helped us beyond words. From understanding him better, how he learns, why he learns a certain way and to help the school understand his potential.
I don't have much time left now but as Annable suggested, I think it wise to choose a psychologist who is experienced with gifted children and who will provide a detailed report.
From the professional advice we have been given to date, one of the main areas to focus on for young gifted children is time on task + effort=achievement. Try not to worry too much about the work being 'too easy' just focus on making sure she completes that work and if she doesn't that there are consequences, this you will need to work on with the school.
It would be ideal if the school can provide work that is above her knowledge base (their duty I know but hard to deliver on it seems). Gifted children need to learn to put in effort on tasks they do not know the answers to (as many of these kids have only been presented with questions they do know the answer to), and to feel failure and then what it means to persist and gain a sense of achievement.
As far as the social side of things, we also had similar problems to you. The advice we were given was to teach very quickly that 'yes, that is the way it is, people are all different...get used to it". Blunt I know, but he/she needs to work through the social scene which is a mine field for any child really. Also model for your daughter lots of social gatherings yourself with a variety of people. Our son still complains about how kids behave but we tell him constantly that not everyone wants to do, be and have what he wants. Finding some like minded friends is crucial too though.
I hope I've helped with my now rushed version of this post. Anything else I can be of help with I'd be happy to respond to. Good luck.