Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7412 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #354 |
| commutator | C2×C42 | C2×C8 |
| Frattini, cosocle | #1728 | C22 |
| socle | C22 | #1319 |
| prime radical | C24 | #100 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #1728 ◁ G | |
| LECS | G ▷ #1728 ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
318 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(2,5,6,7)(9,10,12,15,13,16,11,19,14,18,21,23,22,17,20,24);(1,2,4,6)(3,7,8,5)(9,11,14,20)(10,15)(12,13,21,22)(16,19)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=d,b^2=f,c^2=fi,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=def,e^b=ei,d^c=dhi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1171);