Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149893 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2122 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #13218 |
| prime radical | #24287 | C2 |
| LCS | G ▷ C2×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 451 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,7,14,9,16,25,12)(3,17,19,10,21,11,23,28)(4,22,18,31,20,29,32,27)(5,24,8,13,26,30,6,15);(1,3,8,20,9,21,6,4)(2,10,16,28)(5,23,25,32,26,19,7,18)(11,24,17,30)(12,22,14,29)(13,27,15,31);(1,4,6,21,9,20,8,3)(2,11,13,22,16,17,15,29)(5,23,25,32,26,19,7,18)(10,24,27,14,28,30,31,12)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,f^b=fg,g^b=gh,f^d=fhi,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60109);