Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | D8 |
| commutator | C8 | C82 |
| Frattini, cosocle | C42×C8 | C22 |
| socle | C22 | #1279 |
| prime radical | C23 | #339 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 122 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:
(1,2,3,5,4,6,7,8)(9,10)(11,14)(12,21)(13,17)(15,19)(16,18)(20,23)(22,24);(10,14,16,22,17,23,21,15)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=h,f^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fi,h^a=hi176 conjugacy classes — view the full character table.
| maximal subgroup of | #72216215, #72216228, #72216233, … |
| maximal normal subgroup of | #72216215, #72216228, #72216233, … |
| maximal quotient of | #72216542 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 460);