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×#41979 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5,8,6)(7,9,11,10)(20,21)(22,23)(24,28)(25,26)(27,31)(29,30)(32,33)(34,35);(4,5,8,6)(7,10,11,9)(12,13)(14,15)(16,17)(18,19)(21,26)(22,27)(28,33)(29,34);(5,6)(9,10)(13,17)(14,18)(20,22,25,27)(21,23,26,31)(24,29,32,34)(28,30,33,35);(4,6,8,5)(7,10,11,9)(12,14,16,18)(13,15,17,19)(20,23)(21,27)(22,26)(24,30)(25,31)(28,34)(29,33)(32,35);(4,7)(5,9)(6,10)(8,11)(12,15)(13,18)(14,17)(16,19)(20,24,25,32)(21,28,26,33)(22,29,27,34)(23,30,31,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34967);