Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×C5×#1735 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C5 | D4 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C24×C5 | C2×D4 |
| prime radical | C25×C5 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C5 ◁ C24×C4×C5 ◁ G | |
2398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4)(5,6)(7,8,9,10)(11,12)(13,21)(14,18)(15,19)(16,20)(17,22)(23,26)(24,25);(4,6)(11,13,15,23,16,24,14,22)(12,17,18,25,20,26,19,21);(11,14,16,15)(12,18,20,19)(13,22,24,23)(17,25,26,21)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13333);