Other mistake: The very last scene of the movie shows Supergirl flying underwater holding the Omegadron. Oddly enough, though her body is realistic, her arms are now very thin and measure two meters long.
Supergirl (1984)
1 other mistake
Directed by: Jeannot Szwarc
Starring: Faye Dunaway, Peter O'Toole, Mia Farrow, Hart Bochner, Helen Slater
Continuity mistake: When Supergirl crushes the rock, her hair is curly and messy. In the following shot it's brushed and straight.
Trivia: Helen Slater later played Kara's aunt Lara, the biological mother of her cousin Clark, in Smallville, and Kara's adoptive mother Eliza in the CW's Supergirl.
More questions & answers from Supergirl
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.