Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33588 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 73 points:
(1,2,3,4,5)(7,9)(10,11,16,21)(12,17,23,38)(13,26,29,60)(14,30,34,59)(15,35,32,20)(18,41,43,27)(19,44,46,33)(22,48,49,71)(24,54,52,70)(25,57,53,40)(28,61,36,45)(31,47,62,42)(37,63,64,50)(39,67,66,56)(51,72,58,68)(55,69,73,65);(6,7,8,9)(10,12)(11,17)(13,22)(14,24)(15,25)(16,23)(18,37)(19,39)(20,40)(21,38)(26,48)(27,50)(28,51)(29,49)(30,54)(31,55)(32,53)(33,56)(34,52)(35,57)(36,58)(41,63)(42,65)(43,64)(44,67)(45,68)(46,66)(47,69)(59,70)(60,71)(61,72)(62,73);(10,13,16,29)(11,18,21,43)(12,22,23,49)(14,31,34,62)(15,36,32,28)(17,37,38,64)(19,45,46,61)(20,47,35,42)(24,55,52,73)(25,58,53,51)(26,59,60,30)(27,44,41,33)(39,68,66,72)(40,69,57,65)(48,70,71,54)(50,67,63,56);(12,23)(17,38)(22,49)(24,52)(25,53)(37,64)(39,66)(40,57)(48,71)(50,63)(51,58)(54,70)(55,73)(56,67)(65,69)(68,72)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26576);