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×#41422 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1992 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)(8,9,13,10)(11,16,19,14)(12,15,20,17)(18,22,23,21)(24,25)(26,28)(27,35)(29,30)(31,34)(32,38)(33,39)(36,37);(9,10)(14,16)(15,17)(21,22)(25,30)(28,31)(32,35)(37,39);(4,5,6,7)(8,10,13,9)(11,16,19,14)(12,17,20,15)(18,22,23,21)(24,26)(25,31)(27,36)(28,30)(29,34)(32,37)(33,38)(35,39);(8,11,13,19)(9,14,10,16)(12,18,20,23)(15,21,17,22)(24,27,29,38)(25,32,30,35)(26,33,34,36)(28,37,31,39);(8,12,13,20)(9,15,10,17)(11,18,19,23)(14,21,16,22)(24,28,29,31)(25,26,30,34)(27,37,38,39)(32,33,35,36)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34410);