Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#2207 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
806 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);(4,6)(8,12)(9,15)(10,17)(13,22)(14,21)(16,19)(18,20);(3,4,5,6);(7,8,11,14)(9,16,18,22)(10,19,17,13)(12,20,21,15)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26564);