Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#40578 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1694 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)(4,5,8,9)(6,7,11,10)(20,21,24,22)(23,26,27,25);(5,9)(7,10)(12,13)(14,18)(15,16)(17,19)(20,22,24,21)(23,26,27,25);(5,9)(6,11)(13,16)(17,18)(20,23)(21,25)(22,26)(24,27);(4,6,8,11)(5,10,9,7)(12,13)(14,17)(15,16)(18,19);(4,7,8,10)(5,11,9,6)(12,14)(13,17)(15,19)(16,18)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 33566);