Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131824 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #61482 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
596 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(2,5,6,7)(9,10)(11,23,25,32)(12,17)(13,21)(14,20)(15,22)(16,24,33,39)(18,30)(19,29)(26,37,27,38)(28,35,40,34)(31,36);(1,2)(3,7)(4,6)(5,8)(9,11,15,25)(10,16,22,33)(12,28,31,40)(13,27,14,26)(17,32,36,23)(18,37,19,38)(20,35,21,34)(24,30,39,29);(10,17)(12,29)(13,14)(16,32)(18,20)(19,21)(22,36)(23,33)(24,28)(26,27)(30,31)(34,38)(35,37)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,e^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,h^a=hi,d^b=dh,e^b=egi,f^b=fghi,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39022);