Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140094 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1919 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9538 |
| prime radical | C23 | #443 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
194 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,5,12,24,10,7,13,25,30,9,4,21,23)(3,14,18,29,19,15,31,32,20,22,17,27,16,8,26,28);(1,3,6,18,5,19,24,31,7,20,25,17,9,16,21,26)(2,8,11,28,12,14,10,29,13,15,30,32,4,22,23,27);(1,4,7,12)(2,9,13,5)(3,15)(6,10,25,23)(8,20)(11,24,30,21)(14,16)(17,32)(18,28)(19,22)(26,27)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=dg,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=egi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47290);