Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33522 |
| 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 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
1170 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)(4,5)(6,7)(8,9)(10,15)(11,14)(12,25)(13,18)(16,22)(17,23)(19,29)(20,26)(21,35)(24,28)(27,36)(30,34)(31,33)(32,37)(38,39);(9,14)(11,22)(12,23)(15,26)(16,17)(18,28)(20,34)(21,33)(24,36)(25,27)(29,37)(30,31)(32,39)(35,38);(5,7)(8,10)(9,15)(11,20)(12,21)(13,19)(14,26)(16,30)(17,31)(18,29)(22,34)(23,33)(24,32)(25,35)(27,38)(28,37)(36,39);(10,19)(15,29)(20,32)(21,33)(26,37)(30,38)(31,35)(34,39)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26510);