Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator, prime radical | C24 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | C2×#323 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,7)(4,5)(6,8)(9,10,12,17,13,11,18,20,14,16,22,24,23,19,15,21);(2,5)(6,7)(9,11,12,21,23,10,15,24,14,19,22,20,13,16,18,17);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=e,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,f^b=fg,g^b=gi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49136);