#502658

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61262	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

793 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)(3,7)(4,6)(5,8)(11,21)(12,26)(13,29)(14,28)(16,31)(17,18)(19,30)(22,35)(23,36)(24,25)(27,34)(32,40)(33,39)(37,38);(1,3,4,8)(2,5,6,7)(9,10)(11,22)(12,27)(13,17)(14,30)(15,20)(16,24)(18,29)(19,28)(21,35)(23,39)(25,31)(26,34)(32,38)(33,36)(37,40);(9,11,13,24,14,25,12,21,15,23,26,38,28,37,29,36)(10,16,27,39,19,33,17,31,20,32,18,22,30,35,34,40);(2,6)(5,7)(11,23)(16,32)(21,36)(22,39)(24,38)(25,37)(31,40)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409854);