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×#12995 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1874 |
| prime radical | C5×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
476 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)(6,7,12,19,13,20,37,51,14,21,36,50,11,18,10,17)(8,22,26,44,27,48,24,41,28,55,34,15,25,42,62,43)(9,29,32,49,31,47,35,53,33,52,39,45,38,16,30,46)(23,56,59,67,58,66,57,68,60,69,64,65,63,54,61,40);(6,8,13,25,14,28,11,27)(7,15,20,41,21,44,18,43)(9,23,31,63,33,60,38,58)(10,34,12,24,37,26,36,62)(16,40,29,65,47,68,52,67)(17,48,19,22,51,42,50,55)(30,64,32,57,35,59,39,61)(45,69,46,66,49,56,53,54);(6,9)(7,16)(8,23)(10,35)(11,38)(12,39)(13,31)(14,33)(15,40)(17,49)(18,52)(19,53)(20,29)(21,47)(22,54)(24,61)(25,63)(26,64)(27,58)(28,60)(30,37)(32,36)(34,59)(41,65)(42,69)(43,67)(44,68)(45,51)(46,50)(48,56)(55,66)(57,62)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5983);