Your formula of half- and whole-steps is correct, but the spelling is not.
Different ways of spelling the same pitch are called Enharmonics. You already know that all of the "black key" notes on the piano have more than one name, but you can also alter notes further than a semitone, or make an alteration to two "white keys" that are next to one another.
You are already familiar with
b/# notation: accidentals that modify the pitch by a half step, or semitone. There are also double-sharp and double-flat accidentals that modify by whole steps, or by two semitones.
The reason for using these is to spell every note in the scale with a distinct letter name, and thus, distinct and orderly placement on the musical staff. With enough accidentals, I could spell the "
G#" scale as, say...
G# Bb B# B## Eb E# G G#
But written out on a staff this would be pretty incomprehensible, especially for someone sightreading.
To solve this problem, we start with every note name on the staff:
G A B C D E F G
... and then modify those notes with accidentals to suit the scale in question:
G# A# B# C# D# E# F## G#
Now, this is still a lot of symbology on the page, so to simplify, we use key signatures to put all of this information in one place. Double sharps are almost never used in key signatures, however, so as others have mentioned, the typical spelling of this scale would be its enharmonic equivalent of Ab major (which only has four flats as opposed to 8 sharps).