Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C4 |
| direct product | C3×#12522 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C3 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C82 ◁ G | |
238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3)(4,5,8,11)(6,12,15,19)(7,16,13,10)(9,17,18,14)(20,21)(22,30)(23,26)(24,37)(25,38)(27,33)(28,31)(29,36)(32,42)(34,49)(35,50)(39,47)(40,46)(41,51)(43,45)(44,48);(4,6,13,17,8,15,7,14)(5,9,16,19,11,18,10,12)(21,26)(27,39)(28,40)(29,38)(30,42)(37,50)(41,43)(44,49);(20,22,31,45,25,34,35,47,23,32,46,51,36,48,24,33)(21,27,37,44,29,41,40,42,26,39,50,49,38,43,28,30)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5510);