Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3287 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
650 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)(6,7,9,12)(8,10,13,11)(15,20)(16,18)(21,28)(22,26)(23,27)(24,29);(14,15,19,22)(16,20,18,26)(17,24,25,28)(21,27,29,23);(7,10)(8,13)(11,12)(14,16,19,18)(15,20,22,26)(17,23,25,27)(21,28,29,24);(14,17,19,25)(15,21,22,29)(16,23,18,27)(20,28,26,24)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12758);