Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94642 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1408 |
| commutator | C2×C4×C8 | C2×C4 |
| Frattini, cosocle | #1956 | C22 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1956 ◁ G | |
| LECS | G ▷ #1956 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,11,17,14,20,23,15)(12,18,22,16,13,19,21,24);(2,5)(3,8)(6,7)(10,15)(11,21)(12,13)(16,19)(17,20)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=d,c^2=fi,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gh,h^a=hi,c^b=cf,d^b=defhi,e^b=ehi,f^b=fi,g^b=gi,d^c=dgh,e^d=ehi,f^d=fi,g^d=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1840);