Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C27 ⋊ C2 |
| Sylow | C27 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D27 |
| commutator, cosocle | C2×C27 | C23 |
| Frattini | C2×C9 | C22×D3 |
| Fitting | Q8×C27 | C2 |
| socle | C2×C3 | C22×D9 |
| hypercenter | Q8 | D27 |
| hypocenter | C27 | #41 |
| LCS | G ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
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:
(1,25,24,10,8,5,22,20,23,4,7,26,9,19,14,3,27,6,12,17,13,21,2,18,11,16,15)(28,30,31,34)(29,32,33,35);(28,29,31,33)(30,35,34,32);(2,5)(3,9)(4,12)(6,7)(8,18)(10,11)(13,20)(14,19)(15,25)(16,24)(17,23)(21,22)(26,27)(30,34)(32,35)Polycyclic presentation:
2^4,3^3|b^2=d,c^2=d,e^3=f^2g,f^3=g^2,c^a=cd,e^a=e^2f,f^a=f^2g,g^a=g^2,c^b=cd75 conjugacy classes — view the full character table.
| maximal subgroup of | #11720163, #11720165, #11720167, … |
| maximal normal subgroup of | #11720163, #11720165, #11720167, … |
| maximal quotient of | #11720139, #11720141, #11720142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 50);