Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1033 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1033 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,6)(4,5)(7,8)(9,10,14,13)(11,19,21,16)(12,17,25,24)(15,18,28,27)(20,31,38,37)(22,35,29,32)(23,40,39,33)(26,36,30,34);(2,5)(6,8)(9,11,14,21)(10,16,13,19)(12,23,25,39)(15,29,28,22)(17,33,24,40)(18,35,27,32)(20,30,38,26)(31,36,37,34);(1,3,4,7)(2,6,5,8)(9,12,14,25)(10,17,13,24)(11,20,21,38)(15,30,28,26)(16,31,19,37)(18,36,27,34)(22,39,29,23)(32,40,35,33);(2,5)(6,8)(9,13,14,10)(11,16,21,19)(12,24,25,17)(15,27,28,18)(20,31,38,37)(22,32,29,35)(23,33,39,40)(26,36,30,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407389);