Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1934 |
| prime radical | C23 | #486 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,6,5,9,13,8)(3,10,11,16,12,7,15,14)(18,25)(19,30)(20,21)(22,23)(26,32)(27,29)(28,31);(2,6,9,8)(7,14,10,16)(17,18,26,31,22,28,20,25,24,30,21,27,23,29,32,19);(1,3,4,11,5,12,13,15)(2,7,8,16,9,10,6,14)(17,19,20,28,23,27,26,18,24,25,32,29,22,31,21,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=d,d^2=gi,e^2=hi,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,e^c=egh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47522);