Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
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 | #1925 |
| prime radical | C23 | #440 |
| 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 | |
255 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 86 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)(9,10,14,18,15,20,12,16)(11,21,25,36,22,39,23,38)(13,27,29,37,30,32,26,19)(17,33,34,40,35,28,31,24);(1,3,5,8,4,2,6,7)(10,16,20,18)(11,22)(17,34,35,31)(19,32,37,27)(21,36,39,38)(23,25)(24,40)(28,33);(2,3)(7,8)(9,11,12,23,15,22,14,25)(10,17,18,34,20,35,16,31)(13,28,26,40,30,33,29,24)(19,36,27,39,37,38,32,21)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=d,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^c=egi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47334);