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×#41425 |
| 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 | |
1998 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)(8,9,13,17)(10,18,19,14)(11,15,21,23)(12,16,22,20)(25,29)(26,32)(31,36)(33,38);(10,19)(12,22)(14,18)(16,20)(24,25,28,29)(26,30,32,35)(27,31,34,36)(33,37,38,39);(4,5,6,7)(8,10,13,19)(9,14,17,18)(11,20,21,16)(12,15,22,23)(24,25)(26,30)(27,31)(28,29)(32,35)(33,37)(34,36)(38,39);(8,11)(9,15)(10,16)(12,14)(13,21)(17,23)(18,22)(19,20)(24,26,28,32)(25,30,29,35)(27,33,34,38)(31,37,36,39);(8,12)(9,16)(10,15)(11,14)(13,22)(17,20)(18,21)(19,23)(24,27,28,34)(25,31,29,36)(26,33,32,38)(30,37,35,39)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34413);