Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C25 · Schur multiplier (H2) = C2 |
| direct product | C25×#350 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C25 | D4 |
| commutator | C4 | C2×C8×C25 |
| Frattini, cosocle | C2×C5×C8 | C22×C5 |
| socle | C2×C5 | C5×C22⋊C8 |
| prime radical | C5×#41 | C4×C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C25 ◁ C2×C8×C25 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 2 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)(26,27,30,35,31,36,45,53,32,37,46,54,29,34,43,51)(28,38,40,55,39,49,56,52,41,50,44,57,42,33,47,48);(26,28)(27,33)(29,42)(30,44)(31,39)(32,41)(34,50)(35,52)(36,38)(37,49)(40,46)(43,56)(45,47)(48,54)(51,55)(53,57)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 155);