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) = C23 |
| direct product | C5×#12946 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
402 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)(6,7,9,8)(10,11,15,21)(12,23,26,39)(13,17,31,20)(14,32,28,19)(16,33,30,22)(18,29,37,25)(24,38,41,40)(27,36,34,35);(6,8,9,7)(11,17)(13,28)(14,31)(16,30)(18,35)(19,22)(20,21)(23,38)(24,25)(27,34)(29,41)(32,33)(36,37)(39,40);(10,12)(11,18)(13,29)(14,24)(15,26)(16,34)(17,35)(19,39)(20,36)(21,37)(22,40)(23,32)(25,31)(27,30)(28,41)(33,38)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5934);