Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | #33579 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
387 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 387 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10,12,20,15,22,29,39,14,21,28,18,31,38,13,19)(11,23,24,37,27,36,25,16,26,34,30,17,32,33,40,35);(1,2,6,8,4,7,3,5)(9,11)(10,16,21,35)(12,25)(13,30)(14,26)(15,32)(17,22,37,38)(18,36,20,33)(19,23,39,34)(24,29)(27,31)(28,40);(9,11)(10,17)(12,25)(13,30)(14,26)(15,32)(16,22)(18,34)(19,36)(20,23)(21,37)(24,29)(27,31)(28,40)(33,39)(35,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45820);