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×#35324 |
| 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 | |
1566 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)(5,8)(9,11)(13,14)(17,18)(20,21,25,23)(22,27,30,26)(24,31,32,28)(29,33,35,34);(12,13,16,14)(15,17,19,18)(20,22,25,30)(21,26,23,27)(24,29,32,35)(28,33,31,34);(4,5)(6,9)(7,8)(10,11)(12,14,16,13)(15,18,19,17)(20,23,25,21)(22,26,30,27)(24,31,32,28)(29,33,35,34);(4,6,7,10)(5,9,8,11)(12,15,16,19)(13,17,14,18)(20,22)(21,27)(23,26)(24,29)(25,30)(28,34)(31,33)(32,35);(20,24,25,32)(21,28,23,31)(22,29,30,35)(26,33,27,34)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28312);