Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132201 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#557 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
649 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 649 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(2,5,6,7)(9,10,13,19,14,17,26,33)(11,20,22,35,23,15,28,31)(12,18,25,34,24,16,27,32)(21,29,37,39,36,30,38,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,23)(10,15,17,20)(12,21,24,36)(13,22,26,28)(16,29,18,30)(19,35,33,31)(25,37,27,38)(32,40,34,39);(3,8)(5,7)(10,16)(12,24)(15,29)(17,18)(19,34)(20,30)(21,36)(25,27)(31,40)(32,33)(35,39)(37,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^c=fg,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39435);