Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143606 |
| 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 | #3147 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
381 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(2,6)(3,4)(7,8)(9,10)(11,23)(12,14)(13,31)(15,19)(16,22)(17,21)(18,24)(20,32)(25,38)(26,36)(27,34)(28,37)(29,39)(30,33)(35,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,25)(10,18,12,29)(13,26,32,40)(15,24,33,39)(16,28,17,27)(19,37,30,34)(20,38,31,23)(21,36,22,35);(1,2)(3,6)(4,8)(5,7)(9,12,14,10)(11,23,25,38)(13,31,32,20)(15,19,33,30)(16,21,17,22)(18,24,29,39)(26,35,40,36)(27,34,28,37)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=gi,e^2=gh,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fh,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fh,g^c=gi,h^c=hi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50805);