Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#41443 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5,8,9)(6,7,11,10)(14,18)(16,19)(20,21)(22,24)(23,31)(25,26)(27,29)(28,33)(30,34)(32,35);(5,9)(7,10)(12,13,15,17)(14,16,18,19)(21,26)(22,29)(28,31)(30,35);(5,9)(6,11)(12,14,15,18)(13,16,17,19)(20,22)(21,27)(23,30)(24,26)(25,29)(28,34)(31,32)(33,35);(4,6,8,11)(5,10,9,7)(20,23,25,33)(21,28,26,31)(22,30,29,35)(24,32,27,34);(4,7,8,10)(5,11,9,6)(20,24,25,27)(21,22,26,29)(23,32,33,34)(28,30,31,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34431);