Yes, I felt a little bit how Espergu feels his whole life.
But I did do research, I checked the symbols, I checked the manual.
Is it just a condensator that got fried, I do not see anything wrong (black, smelly) with the whole board.
Check if there is Fuse in the adapter
It's not possible to be just the capacitor (condensator) , there should be transformer from 110V to 5-24v (depend in which mode the GC-100 is working) and also diode (could be Zener one ) which usually get fried if there is no fuse (even if there is they could be fried again) .Those part should be changed with parts like 240V capacitor of same amount of F ,transformer from 240-250v to 5-24v and new diode (because the old one is usually dead) making the circuit as well enough as the one before providing the same output to the GC100.