Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136889 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19133 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
449 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 449 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,8)(9,10,13,18,14,19,30,36,15,17,12,25,27,34,28,38)(11,20,23,33,24,35,29,32,22,39,21,37,31,40,26,16);(3,7)(5,8)(9,11)(10,16,17,32)(12,26)(13,29)(14,31)(15,22)(18,35,25,40)(19,37,34,33)(20,38,39,36)(21,28)(23,30)(24,27);(1,2)(3,8)(4,6)(5,7)(10,17)(12,13)(16,32)(19,34)(21,23)(26,29)(28,30)(33,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ehi,f^c=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44085);