Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
364 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,5,7,4,3,6,8)(9,10,15,23,17,25,13,35,18,26,14,22,16,24,36,21)(11,27,30,40,32,37,34,39,12,33,29,19,31,38,28,20);(1,3,5,8,4,2,6,7)(9,11,17,31,18,12,16,32)(10,19,24,20,26,40,25,39)(13,29,14,34,36,30,15,28)(21,38,22,27,35,37,23,33);(2,3)(7,8)(9,12)(10,20)(11,18)(13,30)(14,28)(15,34)(16,31)(17,32)(19,25)(21,27)(22,37)(23,38)(24,40)(26,39)(29,36)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=eg,f^c=fg,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43927);