Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#61051 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#525 ◁ G | |
2734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,18)(11,15)(12,25)(13,16)(14,21)(17,26)(19,32)(20,29)(22,28)(23,36)(24,30)(27,34)(31,37)(33,39)(35,38);(8,10)(9,14)(11,19)(12,20)(13,22)(15,27)(16,29)(17,28)(18,25)(21,26)(23,33)(24,35)(30,39)(31,38)(32,36)(34,37);(5,7)(8,11,13,24)(9,15,16,30)(10,19,22,35)(12,23,26,37)(14,27,29,39)(17,31,25,36)(18,32,28,38)(20,33,21,34);(10,20)(12,26)(14,28)(17,25)(18,29)(19,33)(21,22)(23,37)(27,38)(31,36)(32,39)(34,35);(9,16)(14,29)(15,30)(17,25)(18,28)(27,39)(31,36)(32,38)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54039);