#502114

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409310);