Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| 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×Q8 | C24 |
| prime radical | #52083 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1047 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1047 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(10,15)(11,19)(13,23)(14,24)(16,32)(20,39)(21,25)(22,38)(28,40)(30,33)(34,36)(35,37);(2,5)(6,8)(9,10)(11,20)(12,15)(13,17)(14,27)(16,22)(18,24)(19,37)(21,35)(23,31)(25,39)(26,36)(28,30)(29,34)(32,40)(33,38);(1,3,4,7)(2,6,5,8)(9,11,12,21)(10,16,15,30)(13,25,23,19)(14,28,24,22)(17,32,31,33)(18,35,27,20)(26,38,29,40)(34,37,36,39);(2,5)(6,8)(10,15)(16,30)(17,31)(18,27)(20,35)(32,33)(34,36)(37,39)Polycyclic presentation:
2^9|c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,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, 407408);