Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C22×C3×#467 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C24×C3×C4 ◁ G | |
3006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(25,26,27);(3,4);(5,6)(7,8)(9,10,13,16)(11,17,20,24)(12,21,18,15)(14,22,23,19);(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17);(6,8)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53424);