Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1229 conjugacy classes of subgroups; only the 292 normal subgroups are shown.
This group has 1229 conjugacy classes of subgroups — too many to draw. Only its 292 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,12,16,13,14,20,21)(11,17,19,24,18,15,22,23);(1,2)(3,6)(4,5)(7,8)(10,14)(12,20)(15,17)(19,22);(2,5)(6,8)(9,11)(10,15,14,17)(12,19)(13,18)(16,23,21,24)(20,22);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=g,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266139);