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×#42899 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1512 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)(12,13,16,17)(14,19,18,15)(21,22)(23,30)(27,29)(31,35);(13,17)(14,18)(20,21,25,22)(23,26,30,28)(24,29,32,27)(31,34,35,33);(4,5)(6,8)(7,9)(10,11)(12,14,16,18)(13,15,17,19)(20,22,25,21)(23,26,30,28)(24,29,32,27)(31,33,35,34);(4,6,7,10)(5,8,9,11)(12,15)(13,18)(14,17)(16,19)(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)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35887);