Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2139 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(23,24,25);(3,4,5,6)(8,12)(9,15)(10,17)(13,20)(14,21)(16,19)(18,22);(7,8)(9,16)(10,19)(11,13)(12,15)(14,17)(18,21)(20,22);(8,13)(12,20)(14,19)(16,21)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26367);