Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138611 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33579 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
377 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10,12,18,14,20,26,15,13,19,28,17,29,36,40,31)(11,21,22,34,25,37,27,33,24,32,23,38,30,35,39,16);(1,2,6,8,4,7,3,5)(10,15,19,31)(12,26)(14,29)(16,21,33,32)(17,20,18,36)(22,27)(23,39)(25,30)(28,40)(34,35,38,37);(9,11)(10,16)(12,27)(13,24)(14,30)(15,21)(17,35)(18,37)(19,33)(20,38)(22,26)(23,40)(25,29)(28,39)(31,32)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45825);