Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#1279 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | D4 |
| commutator | C4 | C2×C3×C4×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle, prime radical | C24×C3 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ C22×C3×C42 ◁ G | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2)(23,24,25);(3,4)(5,6)(8,12)(9,10)(13,14)(15,16,18,21,19,22,17,20);(4,6)(7,8,10,13,11,14,9,12)(16,20,22,21)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 877);