Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17518 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C22×C22⋊C4 |
| prime radical | C5×#3482 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C4×C5×#41 ◁ G | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,10,13,11,8,9,12)(14,15,20,17)(16,23,26,29)(18,22,19,27)(21,28,24,25);(6,8)(7,11)(9,13)(10,12)(14,16)(15,21)(17,24)(18,25)(19,28)(20,26)(22,29)(23,27);(6,9,11,10)(7,12,8,13)(14,17,20,15)(16,24,26,21)(18,22,19,27)(23,25,29,28);(6,9,11,10)(7,12,8,13)(14,18,20,19)(15,22,17,27)(16,25,26,28)(21,29,24,23)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10506);