Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1954 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,7)(3,8)(4,5)(9,10,13,18,15,17,28,30,14,19,27,33,12,26,21,36)(11,20,24,35,25,31,38,39,22,34,37,40,23,16,29,32);(1,2,4,7,6,8,5,3)(9,11,21,29,12,23,27,37,14,22,28,38,15,25,13,24)(10,16,30,35,26,34,18,32,19,31,36,40,17,20,33,39);(1,3,5,8,6,7,4,2)(11,22)(16,31)(20,34)(23,25)(24,37)(29,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=ci104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47903);