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×C4 |
| direct product | C3×#35320 |
| 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 | |
2230 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)(10,14)(12,15)(16,17,21,24)(18,19,22,27)(20,28,29,23)(25,31,30,26);(4,5,6,7)(8,9,11,13)(10,12,14,15)(16,18)(17,19)(20,25)(21,22)(23,26)(24,27)(28,31)(29,30);(8,10,11,14)(9,12,13,15)(16,19)(17,22)(18,24)(20,26)(21,27)(23,30)(25,28)(29,31);(16,19)(17,22)(18,24)(20,26)(21,27)(23,30)(25,28)(29,31);(16,20,21,29)(17,23,24,28)(18,25,22,30)(19,26,27,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28308);