Bel-T is the national block cipher encryption standard of the Republic of Belarus. It operates on 128-bit blocks and uses either 128, 192 or 256-bit keys. Bel-T combines a Feistel network with a Lai-Massey scheme and it has a complex round function with 7 S-box layers. In this work, we use a Mixed Integer Linear Programming (MILP) approach to find a a related-key differential characteristic that extends for 4 rounds and 5 S-box layers ($4 \frac{5}{7}$ rounds) with probability higher than 2^-128. To build an MILP model of Bel-T that a solver can practically handle, we use a partial Difference Distribution Table (DDT) based on the Hamming weight of the input and output differences. The identified differential characteristic is used to mount a key recovery attack on 5 rounds and 6 S-box layers ($5 \frac{6}{7}$ out of 8 rounds) of Bel-T-256 with 2^123.28 chosen plaintexts and 2^228.4 encryptions. According to the best of our knowledge, this is the first public cryptanalysis of Bel-T in the black-box attack model.