Is it permissible to pay a sum of money to transfer to another school that is closer to home?

The basic ruling is that bribery is forbidden and considered one of the major sins, based on the Prophet's (peace and blessings be upon him) saying: "Allah curses the one who gives and the one who takes a bribe." However, this prohibition is specific to bribery used to establish falsehood, invalidate truth, or acquire what one is not entitled to.

As for a bribe given to attain a right or repel injustice or harm, it is permissible according to the preponderant opinion. In this case, the sin falls upon the recipient of the bribe, not the giver. So, if it is your right to transfer to the school but officials refuse unless a bribe is paid, the sin is on them, not on you. However, if it is not your right and you wish to pay a bribe to circumvent conditions, then that is not permissible.