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×#40589 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#548 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2308 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)(16,17,21,22)(18,26,28,23)(19,29,30,24)(20,25,31,27);(8,9)(10,14)(11,12)(13,15)(17,22)(20,31)(23,26)(24,29);(4,5,6,7)(9,12)(13,14)(16,18,21,28)(17,23,22,26)(19,25,30,27)(20,24,31,29);(8,9)(10,13)(11,12)(14,15)(16,19,21,30)(17,24,22,29)(18,27,28,25)(20,26,31,23);(8,10)(9,13)(11,15)(12,14)(16,20)(17,25)(18,24)(19,26)(21,31)(22,27)(23,30)(28,29)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 33577);