Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
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 | #1862 |
| prime radical | C22×D8 | C2×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 | |
327 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 327 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:
(1,2,5,3,4,6,8,7)(9,10)(11,22)(12,27)(13,17)(14,32)(15,21)(16,23)(18,28)(19,30)(20,29)(24,39)(25,34)(26,38)(31,37)(33,35)(36,40);(1,3,5,6,4,7,8,2)(9,11)(10,16)(12,24)(13,31)(14,33)(15,26)(17,34)(18,35)(19,38)(20,39)(21,37)(22,27)(23,28)(25,29)(30,40)(32,36);(10,17)(12,28)(13,30)(14,29)(16,34)(18,21)(19,32)(20,27)(22,39)(23,24)(25,33)(31,40)(35,37)(36,38)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dg,e^b=eg,g^b=gi,d^c=dg,e^c=egi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43081);