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×#2514 |
| 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 | |
1942 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);(4,6)(5,9)(11,12,16,14)(13,18,17,15)(21,25)(23,26);(3,4,7,6)(5,8,9,10)(11,13)(12,15)(14,18)(16,17)(19,20,22,24)(21,26,25,23);(3,5)(4,8)(6,10)(7,9)(11,14,16,12)(13,18,17,15)(19,21,22,25)(20,23,24,26);(3,6,7,4)(5,10,9,8)(11,15,16,18)(12,17,14,13)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27767);