#475704

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

619 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)(11,21)(12,14)(13,27,30,33)(15,19)(16,20)(17,23)(18,28,34,29)(22,31,38,40)(24,32)(25,26)(35,36,39,37);(2,5)(6,8)(9,11,15,24,16,25,14,23)(10,17,12,26,20,32,19,21)(13,22,29,37,30,38,28,36)(18,31,27,39,34,40,33,35);(3,7)(6,8)(9,12,14,20,16,19,15,10)(11,17,23,26,25,32,24,21)(13,18,28,27,30,34,29,33)(22,35,36,31,38,39,37,40);(1,3,4,7)(2,6,5,8)(9,13,14,28,16,30,15,29)(10,18,19,33,20,34,12,27)(11,22,23,36,25,38,24,37)(17,31,21,35,32,40,26,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382900);