If you are living in Germany without a residence permit, that is, you are illegally residing, whether your child can or must attend school depends on your state.  

Be sure to get support from an MBE (Migrationsberatung [Immigration counselling]) office or a family counselling agency. 

Children living illegally in Baden-Wuerttemberg are not obliged to attend school. However, they are allowed. 

Children living illegally in Bavaria are obliged to attend school.

Children living illegally in North Rhine-Westphalia are obliged to attend school.