#501011

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1024 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)(5,6,7,8)(11,20)(16,31)(21,36)(22,24)(23,37)(28,39)(32,33)(35,40);(2,4)(9,10)(11,21)(12,17)(13,18)(14,27)(15,30)(16,23)(19,29)(20,36)(22,35)(24,40)(25,34)(26,38)(28,33)(31,37)(32,39);(6,8)(9,11,13,20)(10,16,18,31)(12,24,15,22)(14,28,26,39)(17,33,30,32)(19,35,34,40)(21,38,36,27)(23,25,37,29);(2,4)(9,12,13,15)(10,17,18,30)(11,22,20,24)(14,25,26,29)(16,32,31,33)(19,27,34,38)(21,35,36,40)(23,39,37,28)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,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, 408207);