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Since you and your child's father have never been married, it is probably unlikely that a court would grant him custody in the absence of other issues (such as if you are an unfit mother). He may, however, be entitled to visitation rights. But, before he can have any visitation rights with your child, paternity must first be established. Unless and until paternity has been established, he has no legal right to see or contact your son.
Here is a link to the Georgia Department of Human Services website that has detailed information on establishing paternity for a child in Georgia:
A child support order will likely be entered for your son if paternity is established. It would be to your son's benefit if his father were to pay child support for him, and it would be his legal obligation as a parent to contribute to his support, but, if you object to him having visitation rights, you would have to appear in court and explain to the judge why you feel that visitation isn't appropriate (basically, not in the best interests of your child). The judge would then decide whether or not to grant visitation rights.
Hope this helps,