Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#12665 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C2×D4 |
| commutator | C4 | C2×C3×C4×C8 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C22×C3 | C8×D4 |
| prime radical | C23×C3 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C22×C3×C16 ◁ G | |
326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(8,9,13,17,12,16,25,11,14,18,30,26,28,35,39,27)(10,19,22,20,23,15,31,32,24,33,29,38,21,34,37,36);(4,5,6,7)(8,10)(9,15)(11,20)(12,23)(13,29)(14,24)(16,33)(17,36)(18,34)(19,35)(21,28)(22,30)(25,37)(26,32)(27,38)(31,39);(8,11,25,35,28,17,13,18,14,27,39,16,12,26,30,9)(10,20,37,19,21,36,29,34,24,38,31,33,23,32,22,15)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5653);