Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C25 · Schur multiplier (H2) = C25 |
| direct product | C25×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C25 | C24 |
| commutator | C2 | C24×C25 |
| Frattini, socle, cosocle | C2×C5 | C24×C5 |
| prime radical | C5×2−1+4 | C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C25 ◁ G | |
219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 4 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)(27,31)(29,37)(32,34)(35,41);(26,27,30,31)(28,34,36,32)(29,33,37,38)(35,40,41,39);(26,28,30,36)(27,32,31,34)(29,35,37,41)(33,39,38,40);(26,29)(27,33)(28,35)(30,37)(31,38)(32,39)(34,40)(36,41)425 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 236);