Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12839 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #427 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1808 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×M16 ◁ C2×C5×#131 ◁ G | |
362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7)(8,18)(9,24,26,14)(10,28)(11,31)(12,16)(13,21)(15,19)(17,29)(20,45,47,38)(22,51)(23,40)(25,55,57,58)(27,36,56,54)(30,59)(32,46,52,68)(33,50)(34,61)(35,43,53,65)(37,44)(39,42)(41,49)(48,63,67,66)(60,69,62,64);(6,8,19,42,11,22,30,50,12,23,44,61,29,49,10,21)(7,13,28,41,17,34,37,40,16,33,59,51,31,39,15,18)(9,20,43,64,27,48,57,68,26,47,65,69,56,67,25,46)(14,32,58,63,36,60,53,45,24,52,55,66,54,62,35,38);(6,9)(7,14)(8,20)(10,25)(11,27)(12,26)(13,32)(15,35)(16,24)(17,36)(18,38)(19,43)(21,46)(22,48)(23,47)(28,58)(29,56)(30,57)(31,54)(33,52)(34,60)(37,53)(39,62)(40,45)(41,63)(42,64)(44,65)(49,67)(50,68)(51,66)(55,59)(61,69)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5827);