Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188766 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,21,15,22,32,40,16,20,33,34,13,19,24,39)(11,23,27,38,28,17,12,29,25,36,31,37,26,35,30,18);(2,5)(3,7)(9,11)(10,17,20,35)(12,24)(13,26)(14,31)(15,28)(16,25)(18,34,29,21)(19,23,22,36)(27,33)(30,32)(37,40,38,39);(1,3,4,7)(2,6,5,8)(9,12,13,27,16,30,15,31)(10,18,19,37,20,29,22,38)(11,24,26,33,25,32,28,14)(17,34,23,40,35,21,36,39);(2,5)(6,8)(11,25)(12,30)(17,35)(18,29)(23,36)(26,28)(27,31)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=ceh,d^b=di,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403839);