Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12483 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1982 |
| prime radical | #19603 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 349 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) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,7)(3,11,14,16,12,15,13,8)(17,18,22,24,31,26,20,30,23,27,32,29,21,19,28,25);(2,7,10,9)(3,12)(8,11,16,15)(13,14)(17,19,21,18,23,26,31,27)(20,30,22,29,28,25,32,24);(1,3,4,13,6,12,5,14)(2,8,9,11,10,16,7,15)(17,20,21,22,23,28,31,32)(18,24,26,30,27,29,19,25)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=e,d^2=ghi,e^2=g,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cg,d^b=dh,f^b=fh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48528);