Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C2×C5×#3127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 47 points:
(1,2)(43,44,45,46,47);(3,4)(5,8)(6,7)(9,10)(11,12,15,19)(13,21,23,17)(14,25,26,34)(16,29,28,20)(18,30,33,27)(22,37,38,42)(24,32,40,36)(31,39,35,41);(4,7)(8,10)(11,13)(12,17)(14,22)(15,23)(16,24)(18,31)(19,21)(20,32)(25,42)(26,38)(27,39)(28,40)(29,36)(30,41)(33,35)(34,37);(11,14,15,26)(12,18,19,33)(13,22,23,38)(16,30,28,27)(17,31,21,35)(20,34,29,25)(24,41,40,39)(32,37,36,42);(3,5,6,9)(4,8,7,10)(13,23)(17,21)(22,38)(24,40)(31,35)(32,36)(37,42)(39,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53544);