Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2,3,4,5)(7,9)(10,11,16,23,40,56,41,22,18,25,14,21,17,24,15,42)(12,26,30,49,44,47,29,19,32,48,28,58,31,55,43,50)(13,33,36,54,45,51,39,53,38,57,34,20,37,52,35,59)(27,61,64,60,68,73,63,69,66,72,62,70,65,46,67,71);(6,7,8,9)(10,12)(11,19)(13,27)(14,29)(15,30)(16,43)(17,44)(18,32)(20,46)(21,55)(22,26)(23,47)(24,58)(25,50)(28,41)(31,40)(33,69)(34,63)(35,64)(36,67)(37,68)(38,66)(39,62)(42,48)(45,65)(49,56)(51,60)(52,70)(53,61)(54,73)(57,71)(59,72);(10,13,18,38)(11,20,25,54)(12,27,32,66)(14,39,16,35)(15,36,41,34)(17,45,40,37)(19,46,50,73)(21,33,23,57)(22,51,42,52)(24,59,56,53)(26,60,48,70)(28,63,30,67)(29,62,43,64)(31,68,44,65)(47,71,55,69)(49,61,58,72)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6008);