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×#42461 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1432 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)(14,18)(16,19)(20,21,25,24)(22,28,30,26)(23,31,33,27)(29,34,32,35);(4,5)(6,10)(7,8)(9,11)(12,13,15,17)(14,16,18,19)(21,24)(26,28)(27,31)(34,35);(5,8)(9,10)(12,14)(13,16)(15,18)(17,19)(20,22,25,30)(21,26,24,28)(23,32,33,29)(27,35,31,34);(4,5)(6,9)(7,8)(10,11)(20,23,25,33)(21,27,24,31)(22,29,30,32)(26,34,28,35);(4,6,7,11)(5,9,8,10)(20,24,25,21)(22,28,30,26)(23,31,33,27)(29,35,32,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35449);