#475170

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,f^d=fi

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, 382366);