#409295

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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)(3,5)(4,6)(7,8)(9,10,15,18,14,19,13,28)(11,20,29,32,23,35,22,16)(12,24,30,38,26,17,27,33)(21,34,40,37,25,39,36,31);(1,3,4,7)(2,5,6,8)(9,11,13,22,14,23,15,29)(10,16,18,20,19,32,28,35)(12,25,27,40,26,21,30,36)(17,31,33,34,24,37,38,39);(9,12,13,27,14,26,15,30)(10,17,18,33,19,24,28,38)(11,21,22,36,23,25,29,40)(16,31,20,34,32,37,35,39);(2,6)(5,8)(12,26)(17,24)(21,25)(27,30)(31,37)(33,38)(34,39)(36,40)

Polycyclic presentation:

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

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, 316491);