Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#41447 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5,9,7)(6,10,11,8)(13,16)(17,19)(21,22)(23,30)(27,29)(31,35);(4,6)(5,10)(7,8)(9,11)(20,21,25,22)(23,26,30,28)(24,29,32,27)(31,34,35,33);(5,7)(6,11)(12,13)(14,17)(15,16)(18,19)(20,22,25,21)(23,26,30,28)(24,29,32,27)(31,33,35,34);(4,7,9,5)(6,8,11,10)(12,14,15,18)(13,17,16,19)(20,23,25,30)(21,26,22,28)(24,31,32,35)(27,33,29,34);(4,8)(5,6)(7,11)(9,10)(20,24)(21,27)(22,29)(23,31)(25,32)(26,33)(28,34)(30,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34435);