Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#12569 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C24×C5 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,21)(13,18)(14,28)(15,31)(16,20)(17,23)(19,29)(22,36)(24,34)(25,40)(26,33)(27,35)(30,41)(32,38)(37,39);(10,12,14,24,13,22,27,39,16,26,30,23,15,25,29,38)(11,17,19,33,31,37,41,36,20,34,35,21,18,32,28,40);(10,13,16,15)(11,18,20,31)(12,22,26,25)(14,27,30,29)(17,32,34,37)(19,28,35,41)(21,36,33,40)(23,38,24,39)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5557);