Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19226 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5)(6,7)(8,15)(9,14)(10,12)(11,18)(13,16)(17,21)(19,20)(22,23,27,31)(24,32,36,46)(25,29,37,43)(26,39,33,30)(28,40,42,35)(34,45,48,53)(38,50,47,44)(41,51,52,49);(7,12)(13,20)(14,18)(15,21)(22,24,33,40,27,36,26,35)(23,28,39,46,31,42,30,32)(25,34,47,51,37,48,38,49)(29,41,50,53,43,52,44,45);(6,8)(7,13)(9,16)(10,17)(11,19)(12,20)(14,21)(15,18)(22,25)(23,29)(24,34)(26,38)(27,37)(28,41)(30,44)(31,43)(32,45)(33,47)(35,49)(36,48)(39,50)(40,51)(42,52)(46,53);(6,9,10,11)(7,14,12,18)(8,16,17,19)(13,21,20,15)(25,37)(29,43)(34,48)(38,47)(41,52)(44,50)(45,53)(49,51)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12214);