Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C22×C3×#397 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25×C3 | C23 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C26×C3 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ G | |
5230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(21,22,23);(3,4);(9,10,11,12)(14,17)(18,19);(6,8)(13,14)(15,19)(16,17)(18,20);(5,6)(7,8)(9,10)(11,12)(13,15,16,20)(14,18,17,19)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27525);