Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9665 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,7,13,22,26,8,14,23,12,6,24,5,11)(3,15,16,28,19,30,17,29,20,25,21,32,18,27,31,10);(2,9,13,26,14,12,24,11)(10,15,28,30,29,25,32,27);(1,3,5,17,6,18,23,16,8,20,22,31,7,19,4,21)(2,10,12,15,13,28,11,30,14,29,9,25,24,32,26,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cgh,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72210674, #72210675, #72320349, … |
| maximal normal subgroup of | #72210674, #72210675, #72320349, … |
| maximal quotient of | #72210510, #72210514, #72210518 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47983);