Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #121 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1797 |
| prime radical | C23 | #407 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,6,4,3,7)(9,10,13,18,14,17,22,24,15,19,21,16,12,23,11,20);(1,3,6,5)(2,7,4,8)(9,11,12,21,15,22,14,13)(10,16,17,20,19,18,23,24);(1,4,6,2)(3,8,5,7)Polycyclic presentation:
2^9|a^2=f,b^2=dg,c^2=g,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,e^c=eg116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41595);