Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#33324 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C2 | C22×C4×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C23×C8 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(10,11,13,18,14,19,23,31,15,17,24,29,25,30,35,39)(12,16,21,27,22,28,34,38,20,26,32,36,33,37,40,41);(6,7,8,9)(10,12)(11,16)(13,21)(14,22)(15,20)(17,26)(18,27)(19,28)(23,34)(24,32)(25,33)(29,36)(30,37)(31,38)(35,40)(39,41);(12,20)(16,26)(21,32)(22,33)(27,36)(28,37)(34,40)(38,41);(11,17)(16,26)(18,29)(19,30)(27,36)(28,37)(31,39)(38,41)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26312);