Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#25983 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15,19,23)(16,25,21,28)(17,26,27,20)(18,24,29,22);(14,16)(15,17)(18,26)(19,21)(20,29)(22,28)(23,27)(24,25);(6,8,9,13)(7,10,11,12)(14,17,19,27)(15,20,23,26)(16,22,21,24)(18,25,29,28);(14,17,19,27)(15,21,23,16)(18,28,29,25)(20,24,26,22)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18971);