Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C26×C32 |
| direct product | C24×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C32 |
| commutator, Frattini, cosocle | C3 | C24×C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(10,12,15)(13,17,16);(3,4)(9,10,13)(11,12,16)(14,15,17);(5,6);(7,8)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 563);