Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144119 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #496 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 416 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)(3,6)(4,8)(5,7)(9,10,16,11)(12,25,27,40)(13,19,32,23)(14,18,33,22)(15,24,31,20)(17,35,21,28)(26,34,36,30)(29,38,39,37);(2,6)(3,5)(7,8)(9,11,16,10)(12,26,27,36)(13,23,32,19)(14,22,33,18)(15,20,31,24)(17,29,21,39)(25,30,40,34)(28,38,35,37);(1,2,4,8)(3,6,5,7)(9,12)(10,17)(11,21)(13,29)(14,28)(15,34)(16,27)(18,36)(19,25)(20,38)(22,26)(23,40)(24,37)(30,31)(32,39)(33,35)Polycyclic presentation:
2^9|a^2=i,b^2=i,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=fghi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fghi,h^b=hi,d^c=dgi,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, 51369);