Continuity mistake: After the hockey ball crashes against Supergirl's back, a girl with glasses staring at her is holding her hockey club. A shot later her arm is raised.
Continuity mistake: When the school bus arrives at the campus there are only 3 or 4 girls inside. From the shots inside Selena's car at least 9 girls leave the bus.
Audio problem: When Nigel holds Linda back after computing class, Lucy mutters "Oh, shoot", but her mouth is saying "Oh, shit."
Continuity mistake: When Supergirl heats the monkey wrench, it swaps positions between shots.
Continuity mistake: At campus, Supergirl holds the road map with her left hand on the side of it. A frame later it lays on the middle of it.
Continuity mistake: When the red haired girl is breaking the pipes, the nerd behind hair has a couple bangs of hair on her forehead, or a lot, depending on the angle.
Continuity mistake: The hole that Supergirl makes in the water tank changes shape between the close and wide shots.
Revealing mistake: When Supergirl is about to fly through the water tank to extinguish the fire, note the patch on the tank where the hole later appears, painted on a darker grey colour.
Revealing mistake: During the bulldozer scene, the truck loaded with hay skids on the road. Skid marks from previous takes are visible, and their colour also changes intensity.
Continuity mistake: When Supergirl runs inside the diner to change into her super outfit, the two girls next to the bathroom door disappear between shots.
Revealing mistake: Right after Supergirl saves Lucy in the bulldozer and dismounts its front part, when she is about to land in the yard to save Ethan, wires holding her are visible.
Revealing mistake: After the bulldozer destroys the gas station, two cars crash against each other. Their hoods are already damaged before the crash.
Continuity mistake: When the bulldozer destroys the gas station's roof, a column falls on top of a green car. Its position changes between shots.
Continuity mistake: When the bulldozer is about to destroy the gas station, two workers next to a green car run away. In the close-up, several seconds later, they're back next to the car, repeating all previous movements.
Continuity mistake: When the bulldozer destroys the bar, wooden planks on top of the door suddenly appear between shots, and the ones to the left of the door disappear. Also, the man on the far left's arms swap from straight to bent between shots.
Continuity mistake: When the crazy bulldozer is about to trap Ethan, check the left side of the street: A woman in yellow stands next to a man with a red jersey, who disappears when the angle changes.
Continuity mistake: Right before the final fight, when Selena is about to say "Power of shadow", smoke comes out behind her. In a wide angle the smoke is gone and it's appearing on the left side first, and later on behind her back.
Revealing mistake: During the beach scene, when Supergirl flies in front of Ethan, the wires holding her are visible.
Revealing mistake: When Kara picks up the Omegahedron, you can briefly see a ring underneath, around her fingers, to make it turn around.
Answer: The Omegaahedron is the power source for the whole city. It supplies the air, the dome that protects them, the lights, and the machines they need to survive.
The question was why would Zoltar need permission to use the Omegahedron, not what it could do.