Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
352 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,5,7,4,3,6,8)(9,10,14,21,16,23,13,18,17,24,12,20,15,22,32,35)(11,25,29,19,26,34,33,39,31,36,27,40,30,37,28,38);(1,3,5,8,4,2,6,7)(10,18)(11,26)(12,13)(14,32)(15,16)(19,36)(20,22)(21,23)(24,35)(25,40)(28,33)(30,31)(34,39)(37,38);(2,3)(7,8)(9,11)(10,19)(12,28)(13,29)(14,33)(15,26)(16,30)(17,31)(18,34)(20,25)(21,36)(22,39)(23,38)(24,40)(27,32)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43924);