Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C52 · Schur multiplier (H2) = C25×C5 |
| direct product | C2×C52×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C52 | C22 |
| commutator, Frattini, cosocle | C2 | C24×C52 |
| socle | C22×C52 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C52 ◁ G | |
656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(11,12,13,14,15);(3,4)(5,8)(6,7)(9,10)(16,17,18,19,20);(4,7)(8,10);(3,5,6,9)(4,8,7,10)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 1188);