Every fingering that fits a hand
0332
open
x776
easy
cbbx
easy
cbba
hard
What is true of it
Gaug is G, B and D♯ — 1, 3 and #5 counted from G.
The easiest of the 4 fingerings below is 0332: 3 fingers, held at the 2nd fret.
1 of them use open strings, which is what makes a shape quiet to change to.
Where its notes are on the neck
The brass dots are G, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
Gaug/B
433x
Gaug/D♯
x332
An inversion is the same chord with a different note lowest. The name changes to a slash name because the bass is what a listener hears as the foundation.