Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12841 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #466 |
| 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 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
358 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)(8,19)(9,26,29,40)(10,30)(11,17)(12,34)(13,18)(14,22)(15,28,39,33)(16,20)(21,48,51,67)(23,44)(24,55)(25,45)(27,38,58,65)(31,41)(32,60)(35,62,63,69)(36,52)(37,53)(42,50,66,54)(43,46)(47,57,61,59)(49,56,68,64);(6,8,20,46,12,24,32,53,13,25,31,52,11,23,10,22)(7,14,30,44,17,36,41,45,18,37,60,55,34,43,16,19)(9,21,47,62,28,50,58,68,29,51,61,69,33,54,27,49)(15,35,59,67,40,64,65,66,39,63,57,48,26,56,38,42);(6,9)(7,15)(8,21)(10,27)(11,33)(12,28)(13,29)(14,35)(16,38)(17,40)(18,39)(19,42)(20,47)(22,49)(23,54)(24,50)(25,51)(26,34)(30,59)(31,61)(32,58)(36,64)(37,63)(41,65)(43,56)(44,67)(45,66)(46,62)(48,55)(52,69)(53,68)(57,60)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5829);