Is a learning center entitled to 25% of the fee of a tutor it nominated to teach a student, then the student's mother changed the student and the educational stage without the center's knowledge, and is the percentage the center takes permissible or impermissible?

It is permissible to agree with the center that they receive 25% of the income from any lesson they nominate for you. This is considered brokerage (samsarah) and a commission (ju'alah), and it is Islamically permissible as long as it does not result in harm. This includes taking a known percentage of the price as a fee for guidance. However, if the mother nominates her other son for study, you are not obligated to pay anything to the center, because they did not nominate him or guide you to him.