diff --git a/gap/byconj.gi b/gap/byconj.gi
index 0d5c554..85ba65e 100644
--- a/gap/byconj.gi
+++ b/gap/byconj.gi
@@ -16,11 +16,23 @@ ConjugatorType@ := obj -> ConjugatorFamily(FamilyObj(obj))!.ConjType;
 
 
 InstallMethod(ConjugatorObj,
-  "for a mult element that allows left quotients (and should be assoc)",
+  "for a mult element that allows quotients (and should be assoc)",
   [IsMultiplicativeElement and IsLeftQuotientElement and IsRightQuotientElement],
   obj -> Objectify(ConjugatorType@(obj), Immutable(obj))
 );
 
+# Even though IsMultiplicativeElementWithInverse implies both
+# IsLeftQuotientElement and IsRightQuotientElement, GAP doesn't seem to
+# calculate these implications on built-in elements. So we need to repeat the
+# method for IsMultiplicativeElementWithInverse, which is true of built-ins,
+# particularly permutations, which serve as a common representation of group
+# elements.
+InstallMethod(ConjugatorObj,
+  "for a mult element with inverse (and should be assoc)",
+  [IsMultiplicativeElementWithInverse],
+  obj -> Objectify(ConjugatorType@(obj), Immutable(obj))
+);
+
 ## Printing and viewing
 InstallMethod(String, "for conjugator objects",
   [IsDefaultConjugatorObject],