Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×#346 |
| prime radical | C24 | M32 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,12,13,16)(8,19)(14,28,26,24)(15,25,27,30)(17,21)(20,32)(29,36,35,34)(31,33);(2,4)(5,6,8,14,9,15,20,29,10,16,21,28,22,30,33,36,11,13,19,26,23,27,32,35,7,12,17,24,18,25,31,34);(6,12,13,16)(8,19)(14,28,26,24)(15,25,27,30)(17,21)(20,32)(29,36,35,34)(31,33)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,f^a=fi,c^b=ch,e^b=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72217181, #72217184, #72217187, … |
| maximal normal subgroup of | #72217181, #72217184, #72217187, … |
| maximal quotient of | #72227023, C2×#151891, #72286359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59087);