Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,22)(13,27)(14,21)(15,20)(16,19)(17,24)(18,29)(23,38)(25,33)(26,34)(28,35)(30,36)(31,37)(32,40)(39,41);(7,9)(10,12)(11,17)(13,28)(14,25)(15,24)(16,26)(18,35)(19,33)(20,34)(21,22)(23,30)(27,38)(29,40)(31,39)(32,37)(36,41);(10,13,16,31)(11,18,19,36)(12,23,26,40)(14,30,15,29)(17,32,33,38)(20,37,21,27)(22,35,34,41)(24,28,25,39);(11,19)(17,33)(18,36)(20,21)(22,34)(27,37)(32,38)(35,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16428);