Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33527 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,19)(11,16)(12,25)(13,23)(14,18)(15,21)(17,27)(20,36)(22,37)(24,34)(26,30)(28,35)(29,33)(31,39)(32,38);(8,10,14,22)(9,15,18,32)(11,20,13,29)(12,26,28,39)(16,30,23,31)(17,33,34,36)(19,35,37,25)(21,24,38,27);(5,7)(8,9)(10,15)(11,23)(12,17)(13,16)(14,18)(19,21)(20,31)(22,32)(24,35)(25,27)(26,33)(28,34)(29,30)(36,39)(37,38);(8,11,14,13)(9,16,18,23)(10,20,22,29)(12,24,28,27)(15,30,32,31)(17,25,34,35)(19,36,37,33)(21,26,38,39)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26515);