You don't mention siblings or close friends so it sounds like she is an only child (your "center of attention" comment) and may have been around a lot of adults. Social skills are not inborn in humans, we have to learn them, and the way to learn is to be around other people.
Try to involve your daughter in settings where she will be around other children and can learn to get along. Gymnastics and computers are wonderful, but they are usually individual efforts. Sign your daughter up for Brownies or a team sport such as soccer. If you have a church or house of worship, encourage her to get involved in group activities there.
People are social beings, and you want your daughter to learn to get along with others. This is an important skill for a 6-year-old as well as a 36-year-old. As she gets older her peers will help her to learn social skills, but those lessons are sometimes painful. Get her started well now so that she will be successful with relationships later.