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×#12926 |
| 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)(10,11,16,22)(12,24,13,27)(14,30,33,39)(15,23,17,21)(18,28,37,25)(19,26,38,29)(20,32,35,34)(31,40,41,36);(6,8,9,7)(10,12,16,13)(11,18,22,37)(14,31,33,41)(15,26,17,29)(19,35,38,20)(21,36,23,40)(24,39,27,30)(25,32,28,34);(10,13,16,12)(11,19,22,38)(14,28,33,25)(15,26,17,29)(18,35,37,20)(21,24,23,27)(30,36,39,40)(31,34,41,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5914);