Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#36670 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| socle | C23×C3 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
1794 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)(12,13)(14,16)(15,17)(18,19)(20,21)(22,26)(23,27)(24,25)(28,31)(29,33)(30,32)(34,35);(4,6,7,10)(5,8,9,11)(14,18)(16,19)(21,25)(26,31)(27,32)(33,35);(5,9)(8,11)(12,14)(13,16)(15,18)(17,19)(20,22)(21,26)(23,29)(24,28)(25,31)(27,33)(30,34)(32,35);(13,17)(14,18)(22,28)(26,31)(29,34)(33,35);(20,23,24,30)(21,27,25,32)(22,29,28,34)(26,33,31,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29658);