Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C25 · Schur multiplier (H2) = C23 |
| direct product | C25×#458 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C25 | #120 |
| commutator | C23 | C23×C25 |
| Frattini, cosocle | C23×C5 | C23×C5 |
| socle | C2×C5 | C5×#120 |
| prime radical | C5×H(5,𝔽2) | C2×C5 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C25 ◁ C23×C25 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,30,33)(28,31,35,38)(29,32,36,39)(34,40,41,37);(26,28)(27,31)(29,34)(30,35)(32,37)(33,38)(36,41)(39,40);(26,29)(27,32)(28,34)(30,36)(31,37)(33,39)(35,41)(38,40)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 959);