For a bipolar transistor to turn on well (good saturation), its datasheet says that its base current should be 1/10th its collector current. Beta is not used for a saturated transistor, beta is used for a linear amplifier that is not saturated and has plenty of collector to emitter voltage. The datasheet shows the range of beta at different collector currents.
A transistor used at its maximum allowed current makes a very lossy switch.
You need to find out the maximum current your motor uses.