Ask about the exhibition space as though you care about what they can offer. Ask about location, what kind of crowds they usually get, the kind of people, etc.

Ask how many illustrations they can put up, obviously.

Ask whether they have any particular preference over what you put up.

Suggest that you'd like to use a certain set of illustrations, because they have a unifying theme or suchlike (so as to seem artistic and professional rather than simply putting up anything to hand).

Ask when it is, how long it will be. Obviously.

After the event, ask whether they'd consider giving you another exhibition, because you have new material that's a bit more experimental. They'll like that and it'll make you look edgy and professional.

And again, obviously. If it goes well, ask if they'd consider giving you work in their company. Their design studio or whatever.

If you have other artistic friends you could even drop their name in the conversation. As a kindly favour. And to make you seem connected.

That what you wanted?