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×#33590 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(5,7)(8,9,14,19)(10,15,22,33)(11,17,13,29)(12,25,28,39)(16,34,36,21)(18,24,37,27)(20,30,23,32)(26,31,38,35);(4,5,6,7)(8,10)(9,15)(11,23)(12,21)(13,20)(14,22)(16,25)(17,32)(18,31)(19,33)(24,38)(26,27)(28,34)(29,30)(35,37)(36,39);(8,10)(9,16)(11,20)(12,26)(13,23)(14,22)(15,25)(17,31)(18,32)(19,36)(21,27)(24,34)(28,38)(29,35)(30,37)(33,39);(8,11,14,13)(9,17,19,29)(10,20,22,23)(12,24,28,27)(15,30,33,32)(16,31,36,35)(18,25,37,39)(21,26,34,38)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26578);