Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8608 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #380 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1797 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 134 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:
(2,5)(3,6)(7,8)(9,10,13,15,12,18,21,24,14,16,22,23,20,17,11,19);(1,2,6,8,4,7,3,5)(9,11,20,22,14,21,12,13)(10,15,18,24,16,23,17,19);(10,16)(11,21)(13,22)(17,18)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ci,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41866);