Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C2×#509 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
387 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 387 conjugacy classes of subgroups — too many to draw. Only its 106 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:
(1,2,4,7,5,3,6,8)(9,10,12,19,15,22,13,20,16,23,32,34,14,21,30,17)(11,24,25,18,28,35,31,37,29,39,40,38,27,33,26,36);(1,3,4,8,5,2,6,7)(10,17)(12,30)(13,32)(14,15)(18,33)(19,21)(20,23)(22,34)(24,36)(25,26)(27,28)(31,40)(35,38)(37,39);(9,11)(10,18)(12,31)(13,25)(14,28)(15,27)(16,29)(17,33)(19,39)(20,35)(21,37)(22,36)(23,38)(24,34)(26,32)(30,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,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=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43889);