Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×C9×S4 (order 864) · Inn(G) = C22 · Out(G) = C22×D3×C9 |
| homology | Abelianization (H1) = C23×C27 · Schur multiplier (H2) = C22 |
| direct product | #41×C27 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C27 | C22 |
| commutator | C2 | C23×C27 |
| Frattini, cosocle | C2×C9 | C23×C3 |
| socle | C2×C3 | C23×C9 |
| prime radical | C3×#41 | C9 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C27 ◁ G | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,11,14,19,16,21,26,12,15,20,17,22,27,24,29,32,18,23,28,25,30,33,31,34,35);(2,5)(6,8);(1,3,4,7)(2,6,5,8)270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 56);