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×#12925 |
| 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 | |
400 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)(12,23)(13,27)(14,21)(15,20)(16,22)(17,24)(18,25)(19,30)(26,38)(28,39)(29,33)(31,37)(32,36)(34,41)(35,40);(6,8,9,7)(10,12)(11,17)(13,28)(14,25)(15,32)(16,26)(18,33)(19,36)(20,34)(21,39)(22,35)(23,27)(24,30)(29,40)(31,41)(37,38);(10,12)(11,18)(13,24)(14,32)(15,25)(16,26)(17,33)(19,35)(20,38)(21,23)(22,36)(27,39)(28,30)(29,41)(31,40)(34,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5913);