Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61852 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C23×C3 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#533 ◁ G | |
1592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 51 points:
(1,2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21,25,29)(22,26,31,40)(23,33,35,43)(24,36,38,28)(27,37,42,34)(30,44,45,51)(32,48,47,41)(39,49,50,46);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17)(20,22)(21,26)(23,30)(24,32)(25,31)(27,39)(28,41)(29,40)(33,44)(34,46)(35,45)(36,48)(37,49)(38,47)(42,50)(43,51);(20,23,25,35)(21,27,29,42)(22,30,31,45)(24,37,38,34)(26,39,40,50)(28,43,36,33)(32,49,47,46)(41,51,48,44);(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18);(5,11)(9,12)(10,19)(17,18)(22,31)(26,40)(30,45)(32,47)(39,50)(41,48)(44,51)(46,49)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54840);