Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143438 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 429 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(2,5)(3,7)(6,8)(9,10,15,20)(11,21,26,39)(12,27,30,37)(13,17,14,19)(16,31,34,23)(18,29,36,28)(22,32,40,33)(24,35,25,38);(3,7)(5,8)(9,11,15,26)(10,16,20,34)(12,22,30,40)(13,25,14,24)(17,32,19,33)(18,35,36,38)(21,37,39,27)(23,29,31,28);(1,2)(3,8)(4,6)(5,7)(10,17)(11,22)(12,28)(13,14)(16,21)(18,36)(19,20)(23,25)(24,31)(26,40)(29,30)(32,35)(33,38)(34,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gi,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fg,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50640);