1) The Pkunk tell you the Ilwrath are "not at home". So expect their homeworld to be outside their sphere of influence.
2) The Ilwrath aren't exactly secretive. Get some ships that can handle playing them and start interrogating...one of them will tell you exactly where their homeworld is - Doggone and Quasar gets them talking
The Ilwrath's homeworld may be pretty far from their sphere of influence on the map, when you first find it. In fact, the system looks alot like a Chmmr zap-sat. Good luck.
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"Even Our House Pets Are Rather Evil." - Ilwrath Captain
I suggest having a paper and pencil handy. They'll throw out a pair of coordinates when you ask them abuot their Doggone Quasars after your third run-in with their fleets.
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We are the Ur-Quan Kohr-Ah. Our nature, the fulfillment of our fate requires your destruction. You are filth. It is now your time to be cleansed.
Seeing as people have been giving you such helpful hints, let me give you a few unhelpful ones, so you have something to chuckle about during your search.
Q: Where is the Ilwrath homeworld?
A: It's in the Ilwrath homesystem. B: It's in Truespace. C: It's close to a star (relatively speaking). D: It's NOT close to a star (relatively speaking). E: It's in this quadrant. F: It's in the immediate vicinity of most Ilwtrath civilians and priets. D: It's not Earth.