Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#33586 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(5,7)(8,9,13,18)(10,14,19,27)(11,15,24,31)(12,25,23,17)(16,32,30,22)(20,26,33,37)(21,36,34,29)(28,39,38,35);(4,5,6,7)(8,10)(9,14)(11,20)(12,21)(13,19)(15,26)(16,28)(17,29)(18,27)(22,35)(23,34)(24,33)(25,36)(30,38)(31,37)(32,39);(9,15)(11,22)(12,23)(14,26)(16,17)(18,31)(20,35)(21,34)(24,32)(25,30)(27,37)(28,29)(33,39)(36,38);(10,19)(14,27)(20,33)(21,34)(26,37)(28,38)(29,36)(35,39)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26574);