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 | #51531 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1028 conjugacy classes of subgroups.
| 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)(9,10,12,15)(11,20,21,35)(13,17,23,31)(14,27,24,18)(16,28,30,22)(19,37,25,39)(26,36,29,34)(32,38,33,40);(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|b^2=i,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, 407410);