Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#35321 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1558 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)(5,6)(9,10)(13,16)(17,19)(20,21,25,23)(22,26,28,29)(24,31,32,27)(30,35,34,33);(4,5,8,6)(7,9,11,10)(12,13)(14,17)(15,16)(18,19)(21,23)(22,28)(27,31)(30,34);(4,6,8,5)(7,10,11,9)(20,22)(21,26)(23,29)(24,30)(25,28)(27,33)(31,35)(32,34);(4,7,8,11)(5,9,6,10)(12,14,15,18)(13,17,16,19)(20,23,25,21)(22,29,28,26)(24,31,32,27)(30,35,34,33);(20,24,25,32)(21,27,23,31)(22,30,28,34)(26,33,29,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28309);