Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#35898 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5,9,7)(6,11,8,10)(14,18)(16,19)(20,21,23,25)(22,26,27,24);(4,6,9,8)(5,10,7,11)(12,13,15,17)(14,16,18,19)(20,21)(22,24)(23,25)(26,27);(4,5,9,7)(6,11,8,10)(12,14)(13,16)(15,18)(17,19)(20,22,23,27)(21,24,25,26);(4,7,9,5)(6,11,8,10);(4,8,9,6)(5,10,7,11)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28886);