Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12298 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1613 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #8854 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,11,13,27)(4,15)(5,19)(6,10)(7,12,22,28)(8,18)(9,21)(14,29,20,24)(16,23,31,26)(17,30)(25,32);(1,3,5,20,6,13,21,14)(2,7,9,26,10,22,19,23)(4,16,18,12,17,31,32,28)(8,24,15,11,25,29,30,27);(4,17)(12,28)(16,31)(18,32)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72217920, #72219578, #72224561, … |
| maximal normal subgroup of | #72217920, #72219578, #72224561, … |
| maximal quotient of | #72217034, #72217037, #72217040 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43210);