Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136733 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 350 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:
(1,2,5,7,4,3,6,8)(9,10,13,18,14,19,30,33,15,20,12,26,29,40,27,16)(11,21,24,39,22,34,31,17,25,35,23,36,32,38,28,37);(1,3,5,8,4,2,6,7)(10,16,20,33)(11,22)(12,27)(13,30)(14,29)(17,34,37,38)(18,19,26,40)(21,39,35,36)(23,24)(25,32);(2,3)(7,8)(9,11)(10,17)(12,28)(13,31)(14,32)(15,25)(16,21)(18,38)(19,39)(20,37)(22,29)(23,27)(24,30)(26,34)(33,35)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ch,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=egi,f^c=fg,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43929);