Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#41411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(5,7)(8,9,13,14)(10,16,18,12)(11,19,22,15)(17,21,20,23)(24,25,29,33)(26,34,35,30)(27,31,37,39)(28,32,38,36);(9,14)(12,16)(15,19)(21,23)(26,35)(28,38)(30,34)(32,36);(4,5,6,7)(8,10)(9,12)(11,20)(13,18)(14,16)(15,23)(17,22)(19,21)(24,26,29,35)(25,30,33,34)(27,36,37,32)(28,31,38,39);(8,11,13,22)(9,15,14,19)(10,17,18,20)(12,21,16,23)(24,27)(25,31)(26,32)(28,30)(29,37)(33,39)(34,38)(35,36);(8,12)(9,10)(11,21)(13,16)(14,18)(15,17)(19,20)(22,23)(24,28)(25,32)(26,31)(27,30)(29,38)(33,36)(34,37)(35,39)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34399);