Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#2097 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C22×C42×C5 ◁ G | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2)(31,32,33,34,35);(7,8,10,12)(9,13,14,11)(15,16,18,22,19,23,27,20)(17,24,25,29,26,30,28,21);(3,4)(5,6)(7,9,10,14)(8,11,12,13)(16,20)(18,27)(21,24)(22,23)(25,28)(29,30);(4,6)(15,17,19,26)(16,21,23,29)(18,28,27,25)(20,24,22,30)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16717);