Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188768 |
| 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 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 417 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(9,10,14,22,15,17,29,33,16,21,30,37,13,20,12,19)(11,23,26,38,27,35,24,39,28,18,31,36,25,32,40,34);(1,2)(3,6)(4,5)(7,8)(10,17,21,20)(12,29)(14,30)(18,32,23,35)(19,37,33,22)(24,40)(26,31)(34,36,39,38);(2,5)(6,8)(9,11)(10,18)(12,24)(13,25)(14,31)(15,27)(16,28)(17,32)(19,34)(20,35)(21,23)(22,38)(26,30)(29,40)(33,39)(36,37);(1,3,4,7)(2,6,5,8)(9,12,13,30,16,29,15,14)(10,19,20,37,21,33,17,22)(11,24,25,26,28,40,27,31)(18,34,35,36,23,39,32,38)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^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,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403850);