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×#12997 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,15,22,14,21,30,18,17,24,13,20,16,23,31,34)(12,25,28,33,32,19,35,37,29,39,26,40,27,38,41,36);(6,7,8,9)(11,18)(13,30)(14,16)(15,31)(19,33)(20,23)(21,22)(24,34)(25,37)(26,35)(27,32)(28,41)(36,39)(38,40);(10,12)(11,19)(13,28)(14,32)(15,26)(16,27)(17,29)(18,33)(20,37)(21,39)(22,36)(23,25)(24,38)(30,41)(31,35)(34,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5985);