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 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 458 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:
(1,2)(3,6)(4,5)(7,8)(9,10,14,20,13,19,28,17,15,18,31,36,29,37,12,26)(11,21,25,32,24,33,27,38,23,39,30,35,40,16,22,34);(2,5)(3,7)(9,11)(10,16,18,33)(12,27)(13,24)(14,30)(15,23)(17,35,26,32)(19,21,37,39)(20,38,36,34)(22,28)(25,31)(29,40);(1,3,4,7)(2,6,5,8)(9,12,13,14,15,28,29,31)(10,17,19,36,18,26,37,20)(11,22,24,25,23,27,40,30)(16,32,21,38,33,35,39,34);(2,5)(6,8)(10,18)(11,23)(17,26)(19,37)(20,36)(22,27)(24,40)(25,30)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,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403838);