Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C3×#34526 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(4,5,7,6)(8,9)(10,16)(11,13)(12,19)(14,17)(15,20)(18,21)(22,23)(24,25,28,31)(26,32,34,29)(27,30,35,37)(33,38,39,36);(9,11)(10,17)(15,19)(18,23)(24,25,28,31)(26,29,34,32)(27,30,35,37)(33,36,39,38);(4,5,7,6)(8,10)(9,14)(11,16)(12,18)(13,17)(15,21)(19,22)(20,23)(24,25)(26,29)(27,30)(28,31)(32,34)(33,36)(35,37)(38,39);(8,11,13,9)(10,16,17,14)(12,19,20,15)(18,22,23,21)(24,26)(25,29)(27,33)(28,34)(30,36)(31,32)(35,39)(37,38);(4,6,7,5)(8,12)(9,15)(10,18)(11,19)(13,20)(14,21)(16,22)(17,23)(24,27,28,35)(25,30,31,37)(26,33,34,39)(29,36,32,38)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27514);