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×#42906 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1748 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)(6,8)(7,10)(9,11)(12,13,16,15)(14,18,19,17)(20,21,25,28)(22,23,26,31)(24,27,33,35)(29,30,32,34);(5,6)(9,10)(20,22,25,26)(21,23,28,31)(24,32,33,29)(27,34,35,30);(4,6,8,5)(7,10,11,9)(12,14,16,19)(13,17,15,18)(20,22)(21,23)(24,29)(25,26)(27,30)(28,31)(32,33)(34,35);(4,7,8,11)(5,9,6,10)(12,15,16,13)(14,18,19,17)(20,23)(21,26)(22,28)(24,30)(25,31)(27,32)(29,35)(33,34);(20,24,25,33)(21,27,28,35)(22,29,26,32)(23,30,31,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35894);