Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C25 · Schur multiplier (H2) = C22 |
| direct product | C25×#137 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C25 | C2×D4 |
| commutator | C4 | C23×C25 |
| Frattini, cosocle | C4×C5 | C23×C5 |
| socle | C2×C5 | C2×C5×D4 |
| prime radical | C5×#41 | C2×C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C25 ◁ C2×C4×C25 ◁ G | |
90 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)(28,34)(29,37)(30,32)(31,35)(33,39)(36,41)(38,40);(26,28,30,36)(27,31,32,40)(29,38,39,35)(33,41,37,34);(27,32)(31,40)(33,37)(34,41)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 198);