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×#41471 |
| 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 | |
1518 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)(13,15)(14,18)(21,25)(26,28)(27,32)(33,34);(12,13,16,15)(14,19,18,17)(21,25)(23,31)(26,28)(29,35);(4,5)(6,8)(7,9)(10,11)(12,14,16,18)(13,17,15,19)(20,21)(22,28)(23,27)(24,25)(26,30)(29,34)(31,32)(33,35);(4,6,7,10)(5,8,9,11)(12,15,16,13)(14,19,18,17)(20,22,24,30)(21,26,25,28)(23,29,31,35)(27,33,32,34);(20,23)(21,27)(22,29)(24,31)(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, 34459);