Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13673 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2196 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #13041 |
| prime radical | #33546 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 567 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,7,12,9,14,6,15)(3,10,13,5,11,8,4,16);(1,3,8,15,9,11,10,12)(2,6,4,5,14,7,13,16);(1,4)(2,10)(3,5)(6,12)(7,15)(8,14)(9,13)(11,16)Polycyclic presentation:
2^9|a^2=f,b^2=df,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=egi,g^a=gh,h^a=hi,e^b=eghi,h^b=hi,d^c=dg,f^c=fg,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,f^e=fhi,g^e=gi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213368, #72284705, #72286053, … |
| maximal normal subgroup of | #72213368, #72284705, #72286053, … |
| maximal quotient of | #72213302, #72241274, #72337618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59752);