Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#2834 |
| 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 | |
1958 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);(4,7)(8,10)(11,12)(13,15)(14,16)(17,18)(19,20,23,25)(21,22,24,26);(5,9)(8,10)(11,12,14,16)(13,17,18,15)(19,21)(20,22)(23,24)(25,26);(3,4,6,7)(5,8,9,10)(11,12)(13,17)(14,16)(15,18)(19,22)(20,24)(21,25)(23,26);(3,5)(4,8)(6,9)(7,10)(11,13)(12,15)(14,18)(16,17)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29644);