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×#61867 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3119 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#533 ◁ G | |
1628 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)(22,27)(23,34)(24,29)(25,38)(26,30)(28,35)(31,46)(32,41)(33,42)(36,44)(37,43)(39,45)(40,49)(47,51)(48,50);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17)(20,22)(21,27)(23,31)(24,37)(25,33)(26,32)(28,40)(29,43)(30,41)(34,46)(35,49)(36,47)(38,42)(39,48)(44,51)(45,50);(20,23)(21,28)(22,31)(24,35)(25,39)(26,36)(27,40)(29,44)(30,45)(32,47)(33,48)(34,38)(37,49)(41,50)(42,46)(43,51);(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18)(22,32)(27,41)(31,47)(33,37)(40,50)(42,43)(46,51)(48,49);(5,11)(9,12)(10,19)(17,18)(20,24,26,25)(21,29,30,38)(22,33,32,37)(23,35,36,39)(27,42,41,43)(28,44,45,34)(31,48,47,49)(40,46,50,51)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54855);