Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139424 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9428 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| prime radical | #3151 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
359 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,6,13,24,14,4,23,7,15,25,10,8,12,5,11)(3,16,20,9,21,26,32,30,22,31,17,27,19,29,18,28);(1,3,8,21,7,22,24,19)(2,9,14,28,15,27,12,30)(4,17,25,32,5,20,6,18)(10,16,23,26,13,31,11,29);(1,4,7,5)(2,10,15,13)(3,17,22,20)(6,8,25,24)(9,16,27,31)(11,14,23,12)(18,21,32,19)(26,30,29,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=gh,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=egi,f^c=fgh,h^c=hi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72231251, #72231260, #72231263, … |
| maximal normal subgroup of | #72231251, #72231260, #72231263, … |
| maximal quotient of | C2×#139425, #72309196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46621);