Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5×C8 · Schur multiplier (H2) = C2×C4×C8 |
| direct product | C5×#7841 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5×C8 | C22 |
| commutator | C2 | C42×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle, prime radical | C23×C5 | C2×C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5×C8 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(18,19,21,24,22,25,28,31)(20,23,27,30,26,29,32,33);(10,11,12,14,13,15,16,17)(18,20)(19,23)(21,27)(22,26)(24,30)(25,29)(28,32)(31,33);(6,7,8,9)(20,26)(23,29)(27,32)(30,33)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 829);