Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 366 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,8)(9,10,14,21,15,18,30,36,16,20,28,38,12,27,13,19)(11,22,26,35,23,37,24,32,25,39,29,17,31,33,40,34);(3,7)(5,8)(9,11,15,31,16,25,12,23)(10,17,18,32,20,35,27,34)(13,29,14,24,30,26,28,40)(19,37,21,22,36,33,38,39);(1,2,4,6)(3,8,7,5)(9,12,16,15)(10,18,20,27)(11,23,25,31)(13,14,30,28)(17,32,35,34)(19,38,36,21)(22,37,39,33)(24,26,40,29)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=h,d^2=gi,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ehi,f^c=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44111);