Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7338 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1316 |
| commutator | C24 | C2×C16 |
| Frattini, cosocle | #2112 | C22 |
| socle | C2 | #7054 |
| prime radical | C25 | C16 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2112 ◁ G | |
| LECS | G ▷ #2112 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
285 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 32 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:
(1,2,3,8,4,9,12,20,5,10,13,21,15,22,25,30,6,7,11,17,14,18,23,27,16,19,24,28,26,29,31,32);(2,7)(3,11)(4,14)(8,17)(10,19)(13,24)(15,26)(21,28)Polycyclic presentation:
2^9|a^2=d,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,d^b=de,f^b=fi,d^c=dg,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 871);