Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138689 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | C2×#2206 | 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 | |
396 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 396 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10,12,17,27,36,13,29,15,20,28,37,14,19,26,18)(11,21,22,38,31,33,40,35,25,39,30,34,24,32,23,16);(1,2,6,8,4,7,3,5)(9,11)(10,16)(12,23)(13,30)(14,31)(15,25)(17,32)(18,21)(19,38)(20,35)(22,26)(24,27)(28,40)(29,39)(33,37)(34,36);(2,7)(5,8)(10,17)(12,26)(13,28)(14,27)(16,32)(18,36)(19,29)(20,37)(21,34)(22,23)(24,31)(30,40)(33,35)(38,39)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,e^a=ei,c^b=cg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45904);