Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23527 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#394 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,20)(10,24)(11,15)(12,18)(13,22)(14,26)(17,31)(19,30)(21,35)(23,29)(25,28)(27,32)(33,36)(34,37);(6,8)(7,12)(9,21)(10,25)(11,19)(13,27)(14,30)(15,28)(16,24)(17,22)(18,26)(20,33)(23,32)(29,35)(31,34)(36,37);(6,9,10,23,11,22,26,37)(7,13,14,29,15,20,24,34)(8,17,18,32,19,21,25,36)(12,27,16,31,28,33,30,35);(9,22)(13,20)(17,21)(23,37)(27,33)(29,34)(31,35)(32,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16515);