Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144122 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #2011 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
389 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,6,3)(4,7,5,8)(9,10)(11,21)(12,27)(13,17)(14,32)(15,20)(16,24)(18,29)(19,28)(22,26)(23,37)(25,38)(30,31)(33,36)(34,39)(35,40);(2,7)(3,8)(4,5)(9,10)(11,22)(12,18)(13,30)(14,32)(15,20)(16,25)(17,31)(19,28)(21,33)(23,36)(24,35)(26,37)(27,29)(34,40)(38,39);(1,3,6,2)(4,8,5,7)(9,11)(10,16)(12,25)(13,24)(14,33)(15,23)(17,36)(18,37)(19,38)(20,34)(21,27)(22,30)(26,28)(29,40)(31,39)(32,35)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=g,e^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fghi,h^b=hi,d^c=dgi,e^c=eh,f^c=fghi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51389);