Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#2049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4,9,14)(5,15,18,30)(6,19,23,28)(7,13,8,12)(10,24,26,17)(11,21,29,22)(16,27,20,25)(31,34,33,32);(3,5)(4,10)(6,20)(7,24)(8,17)(9,18)(11,27)(12,30)(13,15)(14,26)(16,23)(19,33)(21,32)(22,34)(25,29)(28,31);(3,6,7,21,9,23,8,22)(4,11,12,28,14,29,13,19)(5,16,17,32,18,20,24,34)(10,25,15,31,26,27,30,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16512);