Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#2646 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
2374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4,7,8)(5,6,9,10)(12,13)(16,17)(19,20)(21,24)(22,25)(23,26);(4,8)(5,9)(11,12,15,13)(14,16,18,17)(19,21)(20,23)(22,26)(24,25);(3,5)(4,6)(7,9)(8,10)(11,13,15,12)(14,16,18,17)(19,21)(20,24)(22,26)(23,25);(3,6)(4,9)(5,8)(7,10)(11,14)(12,16)(13,17)(15,18)(19,20)(21,23)(22,25)(24,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27552);