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×#41470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2430 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1752 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)(4,5,8,6)(7,10,11,9)(12,13)(14,16)(15,18)(17,19)(21,25)(26,28)(27,32)(33,34);(5,6)(9,10)(13,14)(17,18)(21,25)(23,31)(26,28)(29,35);(4,6,8,5)(7,10,11,9)(12,14,16,13)(15,18,19,17)(20,21)(22,28)(23,27)(24,25)(26,30)(29,34)(31,32)(33,35);(4,7)(5,9)(6,10)(8,11)(12,15,16,19)(13,17,14,18)(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, 34458);