Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C22×C4×C8 | C22 |
| socle | C22 | #1279 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 86 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)(9,10,13,18,15,19,24,17,16,20,12,23,14,11,21,22);(2,5,7,6)(9,11)(10,15)(12,17)(13,22)(14,20)(16,19)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 618);