Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#33584 |
| 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 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
722 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)(10,20)(11,19)(12,15)(13,26)(14,17)(16,23)(18,28)(21,37)(22,36)(24,31)(25,33)(27,35)(29,34)(30,38)(32,39);(4,5,6,7)(9,15)(16,30)(17,19)(18,31)(20,36)(26,33)(32,35)(34,37);(8,10)(9,16)(11,21)(12,22)(13,27)(14,29)(15,30)(17,32)(18,34)(19,35)(20,26)(23,28)(24,38)(25,39)(31,37)(33,36);(8,11,12,14)(9,17,15,19)(10,21,22,29)(13,24,25,28)(16,32,30,35)(18,33,31,26)(20,34,36,37)(23,27,38,39)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26572);