Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| 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 | |
332 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(5,8)(9,10,14,18)(11,19)(12,24,27,38)(13,28,29,17)(15,30)(16,34,36,25)(20,37)(21,32)(22,39)(23,35)(26,31)(33,40);(1,2,4,6)(3,5,7,8)(12,25)(13,29)(16,24)(17,28)(19,37)(20,23)(21,40)(26,30)(27,34)(31,33)(35,39)(36,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,26)(13,20)(14,22)(16,35)(17,31)(18,32)(19,38)(21,34)(23,29)(24,39)(25,30)(27,40)(28,33)(36,37)Polycyclic presentation:
2^9|a^2=fi,b^2=ghi,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=df,e^b=eg,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,h^c=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33415);