Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C25 · Schur multiplier (H2) = C1 |
| direct product | C25×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C25 | D4 |
| commutator | C4 | C2×C4×C25 |
| Frattini, cosocle | C2×C4×C5 | C22×C5 |
| socle | C2×C5 | C5×C4⋊C4 |
| prime radical | C22×C5 | C5×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C25 ◁ C2×C4×C25 ◁ G | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 2 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)(26,27,30,33,31,34,29,39)(28,35,37,40,38,41,36,32);(26,28,29,36,31,38,30,37)(27,32,33,35,34,40,39,41)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 58);