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 | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
606 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 606 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)(3,7)(4,6)(5,8)(9,10,13,20,14,21,29,39,15,19,30,36,12,18,27,35)(11,22,25,38,26,16,31,33,23,17,28,34,24,32,40,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16,19,32)(12,24)(13,28)(14,26)(15,23)(17,18,22,21)(20,37,36,33)(25,30)(27,31)(29,40)(34,39,38,35);(3,8)(5,7)(9,11)(10,17)(12,24)(13,28)(14,26)(15,23)(16,18)(19,22)(20,38)(21,32)(25,30)(27,31)(29,40)(33,39)(34,36)(35,37);(2,6)(5,7)(11,23)(16,32)(17,22)(24,26)(25,28)(31,40)(33,37)(34,38)Polycyclic presentation:
2^9|a^2=g,b^2=f,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, 403825);