Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C5×#60377 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C5×#41 ◁ G | |
4756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,16,20)(12,21,24,30)(13,18,15,28)(14,19,26,34)(17,29,32,22)(23,35,37,40)(25,33,27,38)(31,39,41,36);(10,12)(11,17)(13,22)(14,23)(15,29)(16,24)(18,30)(19,31)(20,32)(21,28)(25,36)(26,37)(27,39)(33,40)(34,41)(35,38);(7,9)(10,13,16,15)(11,18,20,28)(12,22,24,29)(14,25,26,27)(17,30,32,21)(19,33,34,38)(23,36,37,39)(31,40,41,35);(10,14)(11,19)(12,23)(13,25)(15,27)(16,26)(17,31)(18,33)(20,34)(21,35)(22,36)(24,37)(28,38)(29,39)(30,40)(32,41);(14,26)(19,34)(23,37)(25,27)(31,41)(33,38)(35,40)(36,39)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53365);