Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#29742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C24×C4×C5 ◁ G | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(10,11)(12,15)(13,14)(16,17)(18,19,20,24,21,23,25,22);(6,7)(8,9)(11,14)(15,17);(7,9)(10,12)(11,15)(13,16)(14,17)(19,22)(20,25)(23,24);(12,16)(15,17)(19,23)(22,24)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22730);