Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13319 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
572 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)(10,11)(12,23)(13,17)(14,21)(15,20)(16,22)(18,25)(19,27)(24,37)(26,38)(28,35)(29,33)(30,34)(31,36)(32,41)(39,40);(6,7)(8,9)(11,17)(13,27)(14,15)(18,32)(19,21)(20,33)(22,34)(23,37)(24,40)(25,31)(28,41)(29,30)(35,38)(36,39);(7,9)(10,12)(11,18)(13,28)(14,31)(15,25)(16,26)(17,32)(19,37)(20,38)(21,23)(22,36)(24,30)(27,41)(29,40)(33,35)(34,39)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6307);