Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138616 |
| 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 | #33582 | 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 | |
397 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 397 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10,12,18,29,33,13,19,15,21,30,34,14,20,27,16)(11,22,23,17,31,35,28,39,26,38,40,37,25,32,24,36);(1,2,6,8,4,7,3,5)(10,16)(12,27)(13,30)(14,29)(17,32)(18,20)(19,21)(22,36)(23,24)(25,31)(28,40)(33,34)(35,37)(38,39);(9,11)(10,17)(12,28)(13,23)(14,31)(15,26)(16,32)(18,38)(19,35)(20,39)(21,37)(22,34)(24,30)(25,29)(27,40)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45832);