Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C25 · Schur multiplier (H2) = C22 |
| direct product | C25×#504 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C25 | C22 |
| commutator | C2 | C22×C8×C25 |
| Frattini, cosocle | C5×C8 | C23×C5 |
| socle | C2×C5 | C22×C5×C8 |
| prime radical | C5×#41 | C4×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C25 ◁ G | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 3 generators on 57 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)(26,27,29,33,30,34,39,45,31,35,40,46,41,47,51,55)(28,32,37,43,38,44,50,54,36,42,48,52,49,53,56,57);(26,28)(27,32)(29,37)(30,38)(31,36)(33,43)(34,44)(35,42)(39,50)(40,48)(41,49)(45,54)(46,52)(47,53)(51,56)(55,57);(28,36)(32,42)(37,48)(38,49)(43,52)(44,53)(50,56)(54,57)1000 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1005);