Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11059 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
842 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)(7,9)(10,11,15,21,16,19,14,20)(12,17,23,33,24,32,30,37)(13,25,28,35,29,18,27,36)(22,38,26,41,39,31,40,34);(6,7,8,9)(10,12)(11,17)(13,26)(14,30)(15,23)(16,24)(18,34)(19,32)(20,37)(21,33)(22,27)(25,41)(28,39)(29,40)(31,35)(36,38);(10,13,16,29)(11,18,19,25)(12,22,24,39)(14,28,15,27)(17,31,32,38)(20,36,21,35)(23,40,30,26)(33,41,37,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4047);