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×#61856 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#540 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#533 ◁ G | |
1612 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)(22,30)(23,34)(24,28)(25,38)(26,32)(27,36)(29,37)(31,46)(33,41)(35,43)(39,42)(40,50)(44,48)(45,49)(47,51);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17)(20,22,24,32)(21,26,28,30)(23,31,35,47)(25,33,37,49)(27,40,42,44)(29,41,38,45)(34,51,43,46)(36,48,39,50);(20,23,24,35)(21,27,28,42)(22,31,32,47)(25,39,37,36)(26,40,30,44)(29,43,38,34)(33,50,49,48)(41,46,45,51);(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18)(22,32)(26,30)(31,47)(33,49)(40,44)(41,45)(46,51)(48,50);(5,11)(9,12)(10,19)(17,18)(21,28)(22,32)(27,42)(29,38)(31,47)(33,49)(34,43)(48,50)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54844);