Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136735 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,5,7,4,3,6,8)(9,10,15,21,17,22,14,20,18,23,35,36,16,11,13,25)(12,26,30,19,32,37,28,24,33,39,34,27,31,40,29,38);(1,3,5,8,4,2,6,7)(9,11,17,23,18,22,16,10)(12,27,31,38,33,19,32,24)(13,25,15,36,14,20,35,21)(26,28,37,30,39,29,40,34);(2,3)(7,8)(9,12,18,33)(10,19,23,27)(11,24,22,38)(13,34,14,30)(15,28,35,29)(16,32,17,31)(20,37,25,40)(21,39,36,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi,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, 43934);