Hildegard's first appearance is in Episode 21 when she and Fagino are summoned by Rudolph to prevent Noi and the others from escaping the ground, where they had access to important information. To stop them from leaving, they have a big fight, in which Shu and the others are terribly outmatched to these new Investiture Beings and Noi is behind the scenes trying to get the information for the others. In the end, the whole city above is destroyed by Hildegard and Fagino.
Hildegard has two different personalities. They are easy to determine by voice, as one is male and the other is female. The female side of Hildegard is not serious and fools around. She also likes to play games, as show in Episode 21 when instead of referring to her and the others' brawl as a fight, she makes it a game where if any of them touch her and Fagino, then they can pass. This side is more apparent considering Hildegard's appearance, which makes it seem more like Hildegard is a girl.
The male side of Hildegard is almost the opposite. He is much more serious and controlled. He comprises most of Hildegard's personality, but in terms of looks, Hildegard has no reference to being a male whatsoever. However, it is clear enough in appearance in Hildegard's dragon form, which sports two heads that represent the two personalities of Hildegard.
Overall, Hildegard's personality is centered around following the rules and the exact orders of Rudolph. Hildegard has no remorse for trying to kill one of their allies, Rottarace, and ultimately just doesn't care much for the lives of humans, or anyone.