Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×C3×#2508 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4,7,5)(6,9,10,8)(12,14)(13,17)(20,24)(22,25);(4,5)(8,9)(11,12,15,14)(13,16,17,18)(19,20)(21,25)(22,26)(23,24);(3,5,7,4)(6,9,10,8)(11,13)(12,16)(14,18)(15,17)(19,21,23,26)(20,25,24,22);(3,6,7,10)(4,8,5,9)(11,14,15,12)(13,18,17,16)(19,22,23,25)(20,26,24,21)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27766);