Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33520 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
995 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 995 conjugacy classes of subgroups — too many to draw. Only its 164 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 — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,22)(12,16)(13,17)(14,31)(15,32)(18,25)(19,28)(20,27)(21,29)(23,38)(24,37)(26,36)(30,34)(33,35)(39,40);(2,5)(6,8)(10,16)(17,20)(18,34)(19,31)(21,32)(22,38)(35,36)(37,39);(1,3,4,7)(2,6,5,8)(10,17)(13,28)(14,27)(15,29)(16,20)(18,35)(19,21)(22,37)(24,25)(26,33)(30,40)(31,32)(34,36)(38,39);(9,11)(10,18)(12,23)(13,30)(14,24)(15,33)(16,34)(17,35)(19,38)(20,36)(21,39)(22,31)(25,27)(26,29)(28,40)(32,37)Polycyclic presentation:
2^9|c^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,h^a=hi,f^c=fh,g^c=ghi,h^c=hi,f^d=fhi,g^d=ghi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419301);