KöMaL 2016 B4777

There are three species of creatures peopling the planet B-4777: the Alpha, the Beta and the Gamma. People of one species (not necessarily the Alpha) have 2 hands, people of another species have 3 hands, and people of the third species have 4 hands. People of one species (not necessarily those with 2 hands) have 4 fingers on each hand, people of another species have 5, and people of the third species have 6 hands on each hand. Every people represent numbers in a notation with a base equal to the total number of fingers on their hands. (For example, if those with 4 hands have 6 fingers on each, then they will use base-24 representation.) The number $64_{\alpha}$ expressed in the notation of the Alpha people coincides with the number $51_{\beta}$ expressed in the notation of the Beta people. How many hands and how many finger per hand do the Alpha, the Beta and the Gamma have?