Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12738 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
926 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)(6,7)(8,9)(10,11,13,17,15,20,27,19)(12,21,24,35,25,38,22,37)(14,28,26,18,30,34,39,36)(16,29,32,23,33,40,31,41);(10,12)(11,16)(13,22)(14,29)(15,25)(17,31)(18,35)(19,32)(20,33)(21,34)(23,39)(24,27)(26,41)(28,38)(30,40)(36,37);(7,9)(10,13,15,27)(11,17,20,19)(12,22,25,24)(14,26,30,39)(16,31,33,32)(18,34,36,28)(21,37,38,35)(23,29,41,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5726);