#467515

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

616 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)(9,10,13,20,15,19,14,18)(11,16,23,31,22,32,24,36)(12,25,27,35,28,17,29,34)(21,37,26,40,38,30,39,33);(2,5)(6,8)(9,11,13,23,15,22,14,24)(10,16,20,31,19,32,18,36)(12,26,27,38,28,39,29,21)(17,33,34,37,25,40,35,30);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,29)(14,27)(15,28)(16,30)(18,34)(19,25)(20,35)(22,38)(23,39)(24,26)(31,40)(32,37)(33,36);(3,7)(6,8)(10,18,19,20)(11,22)(16,31,32,36)(17,34,25,35)(21,38)(23,24)(26,39)(30,40,37,33)

Polycyclic presentation:

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

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, 374711);