Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145505 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2054 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33819 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,23)(12,17)(13,30)(14,20)(15,32)(16,22)(18,25)(19,28)(21,29)(24,34)(26,40)(27,36)(31,35)(33,39)(37,38);(1,2)(3,7)(4,6)(5,8)(10,17)(11,24)(12,28)(13,14)(15,29)(18,34)(19,21)(20,32)(22,30)(23,36)(25,39)(26,31)(27,38)(35,37);(2,6)(3,4)(7,8)(9,11,12,25,29,39,14,24,16,27,13,31,15,26,28,38)(10,18,19,23,32,37,30,40,22,35,20,36,21,34,17,33)Polycyclic presentation:
2^9|c^2=d,d^2=gi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fhi,g^b=gi,h^b=hi,e^c=egh,f^c=fghi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52701);