#499594

order	512
nilpotency class	4
isoclinism	isoclinic class of #499054
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,14)(13,19)(15,27)(16,24)(17,21)(18,29)(22,30)(23,33)(25,34)(26,37)(28,35)(31,36)(32,39)(38,40);(2,5)(6,8)(9,11,21,26,14,24,13,23)(10,16,17,33,12,20,19,37)(15,30,36,40,29,32,28,25)(18,22,35,38,27,39,31,34);(9,12,14,10)(11,22,24,39)(13,19,21,17)(15,31,29,35)(16,32,20,30)(18,36,27,28)(23,38,26,34)(25,33,40,37);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,17,12,19)(11,23,24,26)(15,28,29,36)(16,33,20,37)(18,35,27,31)(22,38,39,34)(25,32,40,30)

Polycyclic presentation:

2^9|b^2=fh,c^2=i,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406790);