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×#12940 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
424 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)(6,7,9,8)(10,11)(12,23)(13,28)(14,18)(15,33)(16,22)(17,24)(19,29)(20,31)(21,30)(25,40)(26,35)(27,39)(32,38)(34,36)(37,41);(6,8,9,7)(10,12)(11,17)(13,25)(14,32)(15,34)(16,27)(18,35)(19,36)(20,39)(21,40)(22,38)(23,28)(24,29)(26,30)(31,41)(33,37);(11,18)(13,29)(14,31)(15,30)(17,35)(19,22)(20,33)(21,28)(23,40)(24,25)(26,34)(32,41)(36,38)(37,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5928);