Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#35345 |
| 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 | |
1538 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)(4,5,7,9)(6,8,10,11)(13,17)(14,19)(23,30)(27,32)(29,34)(33,35);(12,13,16,17)(14,18,19,15)(21,25)(22,28)(27,32)(29,34);(4,5)(6,8)(7,9)(10,11)(12,14)(13,18)(15,17)(16,19)(20,21,24,25)(22,26,28,31)(23,27,30,32)(29,33,34,35);(4,6,7,10)(5,8,9,11)(12,15,16,18)(13,14,17,19)(20,22)(21,26)(23,29)(24,28)(25,31)(27,33)(30,34)(32,35);(20,23,24,30)(21,27,25,32)(22,29,28,34)(26,33,31,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28333);