Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7339 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1316 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×#407 | C22 |
| socle | C2 | #7187 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ C2×#407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 2 generators on 32 points:
(1,2,5,10,6,9,20,25,7,12,19,28,4,11,18,27)(3,13,15,29,16,23,30,26,17,24,31,32,14,8,21,22);(1,3,4,16,7,17,6,14)(2,8,11,13,12,23,9,24)(5,15,18,30,19,31,20,21)(10,26,27,32,28,22,25,29)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,g^b=gi,d^c=dh,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1279);