Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C82 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7015 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C5 | D4 |
| commutator | C4 | C5×C82 |
| Frattini, cosocle | C43 | C22×C5 |
| socle, prime radical | C23×C5 | #95 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C5 ◁ C43×C5 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5)(6,7)(8,10)(9,12)(11,13)(14,15,16,18,17,19,20,21);(7,10,12,11)(22,23,24,26,25,27,28,29)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3);