Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144131 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 96 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)(9,10,14,20)(11,22,27,35)(12,21,15,18)(13,17,31,36)(16,28,34,24)(19,29,37,30)(23,39,40,38)(25,32,26,33);(2,7)(3,8)(4,5)(9,11)(10,16)(12,28)(13,23)(14,27)(15,24)(17,33)(18,35)(19,38)(20,34)(21,22)(25,29)(26,30)(31,40)(32,36)(37,39);(1,3,6,2)(4,8,5,7)(10,17)(11,23)(12,15)(13,30)(18,37)(19,21)(20,36)(22,35)(24,25)(26,28)(27,40)(29,31)(32,39)(33,38)Polycyclic presentation:
2^9|a^2=g,c^2=gi,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=fghi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fghi,h^b=hi,d^c=dg,e^c=eh,f^c=fghi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51383);