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×#40607 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2376 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)(16,17,21,20)(18,24,26,22)(19,27,28,23)(25,29,31,30);(8,9)(10,14)(11,12)(13,15)(16,18,21,26)(17,22,20,24)(19,25,28,31)(23,29,27,30);(4,5,6,7)(9,12)(13,14)(16,18)(17,22)(19,25)(20,24)(21,26)(23,29)(27,30)(28,31);(8,9)(10,13)(11,12)(14,15)(16,19,21,28)(17,23,20,27)(18,25,26,31)(22,29,24,30);(8,10)(9,13)(11,15)(12,14)(16,20,21,17)(18,24,26,22)(19,27,28,23)(25,30,31,29)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 33595);