Hello bubuchacha,
You had better find out exactly what the law in both countries requires you to do in order to take a minor child out of the country you are now in and German law regarding entry of a minor child or you could wind up with more headaches that you can possibly imagine.
It's a very strictly controlled process nowadays for a minor to travel in the company of one parent only, what with international child abduction and trafficking in human beings being what they are, almost every country in the world has signed the Hague Convention on the Civil Aspects of International Child Abduction. They have also since adopted their own very stringent regulations regarding immigration of minor children with a single parent. Unless they have been revoked by the courts, the fact that you and the child's father never married, have no bearing whatsoever on his parental rights. It is more complicated still if there are no legal arrangements regarding custody of the child, which is one thing that many single mothers mistakenly believe. They assume that since there is no marriage the father's rights do not exist, this is completely incorrect. Parental rights can only be removed by the courts.
Generally speaking, unless you have a court order granting you full custody of the child, you are either going to have the father's notarized permission or a court order to satisfy immigrations authorities in both countries. You may even require one or the other even if you do have full custody depending on local laws since legally the issue of child custody is NEVER closed and can be altered at any time. If the whereabouts of the child's father are unknown then it is quite likely you are going to need to apply to the courts for an order granting permission to remove your child from the country.
Another thing that you must also consider is that even if you do have full custody, the planned move outside the country (which could make visitation extremely difficult) can be used as legal grounds to apply for a change of custody, this is something that can provoke even the most remote father to seek to gain custody as opposed to losing contact altogether, so it shouldn't come as a big surprise to you. This is also a factor that the courts will take a serious look at before granting an order to permit the move, even if the child's father has not been paying support, the two issues (support & visitation) are legally separate issues and have nothing to do with eachother. Even a deadbeat dad can prevent an international move if he so desires.
If you don't have either one of those and you try to leave the country you could face serious criminal charges. Better safe than sorry, it doesn't cost anything other than a few phone calls to the immigrations authorities in both countries to find out just what the legal requirements are. And depending on the situation between you and the child's father (war or peace) it doesn't take more than a few phone calls from him to the immigrations authorities to throw up all kinds of roadblocks that will make your planned trip seem like the Quest for the Holy Grail.
Cheers,
William James Woodward Brazil Animator, Expat-blog Team