Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C33 |
| direct product | C33×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C33 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C33 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C33×C4 ◁ G | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(10,11,12,13,14,15,16,17)(18,25,24,23,22,21,20,19);(4,5,6)(10,18,14,22)(11,19,15,23)(12,20,16,24)(13,21,17,25);(7,8,9)189 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 519);