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 | C22×D8 | 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 | |
360 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 360 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,3,6,8)(9,10,13,19,16,21,14,35,17,22,34,25,15,11,23,20)(12,26,28,18,31,37,29,24,32,39,36,27,30,40,33,38);(1,3,5,8,4,2,6,7)(9,11)(10,15)(12,27)(13,25)(14,35)(16,22)(17,21)(18,32)(19,34)(20,23)(24,31)(26,36)(28,39)(29,37)(30,38)(33,40);(2,3)(7,8)(9,12,17,32)(10,18,22,27)(11,24,21,38)(13,33,34,29)(14,36,23,28)(15,31,16,30)(19,26,25,39)(20,37,35,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=i,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ch,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=egi,f^c=fg,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, 43920);