Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#36676 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| socle | C23×C3 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
2006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5,7,8)(6,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21,23,25)(22,26,27,24);(5,8)(9,11)(20,22)(21,24)(23,27)(25,26);(4,6)(5,9)(7,10)(8,11)(13,16)(17,19)(22,27)(24,26);(6,10)(9,11)(21,25)(24,26);(12,14,15,18)(13,17,16,19)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29664);