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×#26920 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#2011 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,9,10)(8,12,13,11)(14,15,17,20,19,23,18,22)(16,24,25,29,27,28,26,21);(7,10)(11,12)(14,15)(16,21)(17,22)(18,20)(19,23)(24,26)(25,28)(27,29);(15,20)(17,18)(21,28)(22,23)(24,29)(25,26);(6,8,9,13)(7,11,10,12)(14,16)(15,21)(17,26)(18,25)(19,27)(20,28)(22,24)(23,29)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19908);