Ahh ok, maybe she isn't being challenged as far as her school work? The personal space thing (being too close while talking) I see that a lot with some children, I think its just a phase.
Ask her about her work in class, see if she is bored. My 6 year old would act up in class during kindergarten during work time or "story" time and he told us he was always bored. We looked at his school work and he was being taught things he knew how to do since pre-school. Your daughter may be bored with her work and looking for other things to do. How is her school work? Does the teacher have to insist that she completes it?
I'd venture to say your daughter is fine and dandy :) Just try to investigate a bit more of the classroom setting, maybe sit in the class one day (if you can afford the day off work, etc) and see how much of a difference the class/school is to home. As far as the other kids pushing her around, that will be a larger task to handle. You have to find a middle ground with telling her not to let kids treat her like that, but also letting her know not to act out or be mean to them because of their actions.
She is an only child. She does gymnastics in a group of girls age 6-10 twice a week, attends Sunday school, has been in preschool and daycare. My mom says she often gets "run over" by the bigger kids on the rings or monkey bars (kicking, and pushing her off) and she cries or sulks. Her just says she is less mature than the rest of the class. She is outgoing, makes friends with kids of all ages wherever we go. We take her to lots of kid oriented evernts. I have noticed she does tend to get in a little close to people when she talks to them and she is a bit of a clown. In Kindergarten she was praised for being helpful and kind to the other students. I am concerned that now she is in a grade 1/2 split she is being expected to fit in at a grade 2 level. She does like it when the bigger kids give her a hug and attention but does not act infantile.
Is she an only child? How much interaction does she get with kids outside of school? If she is an only child, it may be because she is trying to get attention from other children and she is trying to make friends. Is she just a class clown?
It may just mean she has to take some time to socialize, maybe try getting her into a soccer or t-ball league.
What type of immaturity is she displaying? Just trying to have fun or acting like an infant? Does the teacher or other kids react and entertain her immaturity?