Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
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 | #3169 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
371 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 371 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:
(1,2,3,7,5,8,4,6)(9,10,15,18,14,20,32,38,13,19,31,22,30,17,35,36)(11,21,24,37,23,27,16,33,25,12,26,28,39,29,34,40);(2,6)(3,4)(7,8)(9,11,15,26,14,23,32,34,13,25,31,24,30,39,35,16)(10,12,22,28,20,29,36,40,19,21,18,37,17,27,38,33);(2,7)(3,4)(6,8)(9,12)(10,16)(11,22)(13,21)(14,29)(15,28)(17,24)(18,25)(19,34)(20,26)(23,36)(27,30)(31,37)(32,40)(33,35)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,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, 43872);