Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125647 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #31071 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,16,22)(11,23,28,39)(12,18,30,19)(13,31,29,37)(14,21,15,20)(17,24,36,25)(26,33,27,38)(32,35,40,34);(2,5)(3,8)(6,7)(9,11)(10,17)(12,24)(13,32)(14,27)(15,26)(16,28)(18,33)(19,38)(20,35)(21,34)(22,36)(23,31)(25,30)(29,40)(37,39);(9,12,16,30)(10,18,22,19)(11,24,28,25)(13,14,29,15)(17,33,36,38)(20,31,21,37)(23,34,39,35)(26,32,27,40)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,d^b=df,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32846);