Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#42481 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(5,7)(16,17,22,27)(18,24,19,23)(20,21,26,25)(28,29,30,31);(8,9)(10,14)(11,12)(13,15)(16,18,22,19)(17,23,27,24)(20,28,26,30)(21,31,25,29);(4,5,6,7)(9,12)(13,14)(16,19,22,18)(17,24,27,23)(20,28,26,30)(21,31,25,29);(8,9)(10,13)(11,12)(14,15)(16,20,22,26)(17,25,27,21)(18,28,19,30)(23,29,24,31);(8,10,11,15)(9,13,12,14)(16,21,22,25)(17,26,27,20)(18,29,19,31)(23,28,24,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35469);