Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12483 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | |
370 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 90 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)(9,10,14,19,16,22,32,39,15,21,12,29,31,34,13,20)(11,23,26,37,28,35,25,36,27,40,24,18,33,38,30,17);(1,2,6,8,4,7,3,5)(9,11,15,27)(10,17)(12,30,14,25)(13,24,32,26)(16,33,31,28)(18,22)(19,35)(20,40)(21,36)(23,39)(29,38)(34,37);(2,7)(5,8)(9,11)(10,18)(12,30)(13,24)(14,25)(15,27)(16,33)(17,34)(19,23)(20,35)(21,37)(22,36)(26,32)(28,31)(29,40)(38,39)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,c^b=cg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45898);