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×#33585 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
748 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)(10,14)(11,15)(12,25)(13,18)(16,22)(17,23)(19,27)(20,26)(21,35)(24,31)(28,34)(29,33)(30,36)(32,37)(38,39);(4,5,6,7)(8,10)(9,14)(11,20)(12,21)(13,19)(15,26)(16,28)(17,29)(18,27)(22,34)(23,33)(24,32)(25,35)(30,38)(31,37)(36,39);(9,15)(11,22)(12,23)(14,26)(16,17)(18,31)(20,34)(21,33)(24,36)(25,30)(27,37)(28,29)(32,39)(35,38);(10,19)(14,27)(20,32)(21,33)(26,37)(28,38)(29,35)(34,39)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26573);