Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136906 |
| 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 | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 427 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:
(1,2,3,6,4,7,8,5)(9,10,14,20,15,21,13,33,17,23,12,31,16,22,32,18)(11,24,27,19,28,35,26,36,30,40,25,38,29,39,34,37);(2,5)(3,8)(6,7)(10,18,23,33)(12,32)(13,14)(15,16)(19,35,38,39)(20,21,31,22)(24,37,40,36)(25,34)(26,27)(28,29);(9,11,17,30)(10,19,23,38)(12,26,14,34)(13,25,32,27)(15,29,16,28)(18,35,33,39)(20,24,31,40)(21,37,22,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309441, #72309463, #72309469, … |
| maximal normal subgroup of | #72309441, #72309463, #72309469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44103);