Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,5,8,6,3,4,7)(9,10,15,19,16,18,34,37,13,20,30,22,14,31,33,39)(11,21,24,38,26,32,36,40,25,12,27,29,23,28,17,35);(1,3,6,2)(4,7,5,8)(9,11,15,27,16,26,34,17,13,25,30,24,14,23,33,36)(10,12,22,29,18,28,39,35,20,21,19,38,31,32,37,40);(1,2,6,3)(4,8,5,7)(9,12,13,21)(10,17,20,36)(11,22,25,19)(14,32,16,28)(15,29,30,38)(18,24,31,27)(23,37,26,39)(33,40,34,35)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eg,f^c=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43914);