Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61787 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3547 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
1682 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)(20,21,25,30)(22,26,32,39)(23,28,34,42)(24,27,36,29)(31,38,45,50)(33,37,46,40)(35,41,47,43)(44,48,51,49);(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18)(20,22,25,32)(21,26,30,39)(23,31,34,45)(24,33,36,46)(27,37,29,40)(28,38,42,50)(35,44,47,51)(41,48,43,49);(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(21,27)(24,36)(26,37)(28,41)(29,30)(33,46)(35,47)(38,48)(39,40)(42,43)(44,51)(49,50);(6,14)(9,12)(13,19)(16,18)(20,23)(21,28)(22,31)(24,35)(25,34)(26,38)(27,41)(29,43)(30,42)(32,45)(33,44)(36,47)(37,48)(39,50)(40,49)(46,51);(5,11)(9,12)(10,17)(16,18)(22,32)(23,34)(26,39)(28,42)(33,46)(35,47)(37,40)(41,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54775);