Continuity mistake: After her fight with the invisible monster, Supergirl walks by the broken steel door she fell against, and it looks more crumpled and worn out than before.
Continuity mistake: In the girls' dorm, when Supergirl is trying the bra on, the teddy bear in the back changes position between shots.
Continuity mistake: Supergirl crushes a rock and gets dirt all over the sleeve of her suit. A second later dirt is gone and the suit is clean.
Continuity mistake: Selena starts to say that everything is history and turns around to say the world is hers. Right then, angle changes and cupboards appear on the right from nowhere. When the angle changes, the cupboards disappear.
Continuity mistake: Selena says "Bad luck" and places her hand on the fridge door. In the next shot, only the tips of her fingers are on it.
Revealing mistake: When Supergirl falls on the planet ground and gets released from the mirror trap, she gets up and the hard floor stretches down, revealing it's a rubber mat to soften her fall.
Continuity mistake: While Selena is burning the chicken, in the close up shot big flames come out, but a split second later, from a different angle, there's just smoke.
Continuity mistake: When Supergirl enters her dorm, the Cornflakes box changes position from standing upright to sideways.
Continuity mistake: At Selena's picnic, the label of the champagne keeps turning to face the camera.
Continuity mistake: When at home, Selena says that the world is at her wish and looks down at the bronze sculpture which she is holding close to her chin. In the following angle her hand swaps to stomach level.
Continuity mistake: Supergirl's bracelet swaps wrists while she's flying at night.
Continuity mistake: When Selena is holding the glass champagne and says "I can't wait 'till the world is mine", from a back shot her finger is below liquid level, in the following shot it is above it.
Revealing mistake: When Selena uses her magical powers to accelerate the fairground ride to superfast speed, the twinkling fairground lights in the background are also flashing much faster. They obviously played the tape back at high speed to achieve this.
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.