Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#24262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(7,8)(10,11)(12,15)(13,16)(14,17)(18,23)(19,22)(20,24)(21,25);(6,7,9,8)(10,12)(11,15)(13,18)(14,19)(16,23)(17,22)(20,25)(21,24);(10,13)(11,16)(12,18)(14,20)(15,21)(17,24)(19,25)(22,23);(6,8,9,7)(13,20)(16,24)(18,25)(21,23)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17250);