Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C25 · Schur multiplier (H2) = C22 |
| direct product | C25×#430 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C25 | C2×D4 |
| commutator | C4 | C22×C4×C25 |
| Frattini, cosocle | C2×C4×C5 | C23×C5 |
| socle | C2×C5 | C2×C5×C4⋊C4 |
| prime radical | C23×C5 | C5×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C25 ◁ C22×C4×C25 ◁ G | |
153 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)(26,27,30,33,31,34,29,39)(28,35,38,40,36,41,37,32);(26,28,29,37,31,36,30,38)(27,32,33,35,34,40,39,41);(28,36)(32,40)(35,41)(37,38)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 931);