Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94303 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1393 |
| commutator | C2×M16 | C42 |
| Frattini, cosocle | C2×#407 | C22 |
| socle | C2 | #7250 |
| prime radical | C25 | Q16 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ C2×#407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 372 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 — 2 generators on 32 points:
(1,2,5,10,7,11,6,22)(3,12,14,29,16,30,15,26)(4,17,19,28,21,27,20,9)(8,18,23,13,25,31,24,32);(1,3,5,15,7,16,6,14)(2,8,10,24,11,25,22,23)(4,18,20,13,21,31,19,32)(9,26,17,30,28,29,27,12)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,g^b=gh,e^c=ehi,g^c=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1502);