Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#52015 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1998 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)(4,5,6,7)(17,21)(19,27)(22,24)(25,31);(8,9)(10,12)(11,14)(13,15)(16,17,20,21)(18,22,26,24)(19,28,27,23)(25,30,31,29);(4,5)(6,7)(9,10)(13,14)(16,17)(18,24)(19,23)(20,21)(22,26)(25,30)(27,28)(29,31);(8,10,12,9)(11,14,15,13)(16,18)(17,22)(19,25)(20,26)(21,24)(23,29)(27,31)(28,30);(8,11,12,15)(9,13,10,14)(16,19,20,27)(17,23,21,28)(18,25,26,31)(22,29,24,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 45003);