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×#34530 |
| 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)(10,14)(11,18)(12,13)(15,19)(16,22)(17,20)(21,23)(24,25,29,32)(26,27,34,30)(28,36,37,31)(33,38,39,35);(9,13)(14,20)(15,18)(21,22)(24,26,29,34)(25,30,32,27)(28,33,37,39)(31,38,36,35);(4,5,7,6)(8,10,12,17)(9,14,13,20)(11,16,19,23)(15,21,18,22)(24,26)(25,30)(27,32)(28,33)(29,34)(31,38)(35,36)(37,39);(8,9)(10,14)(11,15)(12,13)(16,21)(17,20)(18,19)(22,23)(24,27,29,30)(25,26,32,34)(28,35,37,38)(31,33,36,39);(4,6,7,5)(8,11)(9,15)(10,16)(12,19)(13,18)(14,21)(17,23)(20,22)(24,28,29,37)(25,31,32,36)(26,33,34,39)(27,35,30,38)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27518);