Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#2606 |
| 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);(11,12,15,14)(13,17,18,16)(20,21)(22,26);(4,7)(8,10)(13,18)(16,17)(19,20,23,21)(22,24,26,25);(3,4)(5,8)(6,7)(9,10)(11,13,15,18)(12,16,14,17)(19,21,23,20)(22,25,26,24);(3,5,6,9)(4,8,7,10)(11,14,15,12)(13,17,18,16)(19,22)(20,24)(21,25)(23,26)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27850);