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×#2493 |
| 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 | |
2398 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);(3,4,7,5)(6,9,10,8)(11,12,15,18)(13,17,14,16)(20,23)(24,26);(4,5)(8,9)(11,13,15,14)(12,16,18,17)(19,20)(21,24)(22,23)(25,26);(3,5,7,4)(6,9,10,8)(11,14,15,13)(12,17,18,16)(19,21,22,25)(20,24,23,26);(3,6)(4,8)(5,9)(7,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,24)(22,23)(25,26)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27737);