Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #3497 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
900 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 900 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,19)(11,16,13,27)(12,22,26,38)(14,29,30,18)(17,20,34,24)(21,35,28,31)(23,37,39,33)(25,32,36,40);(2,5)(6,7)(9,11,15,13)(10,16,19,27)(12,23,26,39)(14,28,30,21)(17,32,34,40)(18,35,29,31)(20,36,24,25)(22,37,38,33);(9,12,13,24,15,26,11,20)(10,17,16,22,19,34,27,38)(14,25,21,23,30,36,28,39)(18,33,35,40,29,37,31,32);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410001);