Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61877 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#533 ◁ G | |
1600 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)(20,21,26,29)(22,27,32,43)(23,34,36,45)(24,30,25,38)(28,39,44,35)(31,46,47,51)(33,41,37,42)(40,48,50,49);(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)(30,42)(34,46)(35,49)(36,47)(38,41)(39,48)(44,50)(45,51);(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)(41,46)(42,51)(43,50);(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18)(21,29)(22,32)(28,44)(30,38)(31,47)(33,37)(34,45)(48,49);(5,11)(9,12)(10,19)(17,18)(20,24,26,25)(21,30,29,38)(22,33,32,37)(23,35,36,39)(27,41,43,42)(28,45,44,34)(31,48,47,49)(40,46,50,51)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54865);