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×#35323 |
| 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 | |
1558 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)(5,8)(9,11)(13,17)(14,19)(20,21,25,22)(23,26,30,28)(24,29,32,27)(31,34,35,33);(12,13)(14,18)(15,19)(16,17)(21,22)(23,30)(27,29)(31,35);(4,5)(6,9)(7,8)(10,11)(12,14,16,19)(13,18,17,15)(20,22,25,21)(23,28,30,26)(24,29,32,27)(31,34,35,33);(4,6,7,10)(5,9,8,11)(12,15,16,18)(13,19,17,14)(20,23)(21,26)(22,28)(24,31)(25,30)(27,33)(29,34)(32,35);(20,24,25,32)(21,27,22,29)(23,31,30,35)(26,33,28,34)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28311);