Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#61097 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
2654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,18)(11,15)(12,25)(13,16)(14,21)(17,28)(19,36)(20,34)(22,39)(23,35)(24,32)(26,30)(27,33)(29,38)(31,37);(8,10,13,20)(9,14,16,30)(11,22,24,38)(12,26,28,21)(15,31,32,36)(17,34,25,18)(19,27,37,23)(29,35,39,33);(5,7)(8,9)(10,14)(11,15)(12,17)(13,16)(18,21)(19,29)(20,30)(22,31)(23,33)(24,32)(25,28)(26,34)(27,35)(36,38)(37,39);(8,11,13,24)(9,15,16,32)(10,19,20,37)(12,27,28,23)(14,29,30,39)(17,35,25,33)(18,36,34,31)(21,38,26,22);(9,16)(10,20)(15,32)(17,25)(19,37)(21,26)(22,38)(33,35)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54085);