#470741

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

753 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)(3,5)(4,6)(7,8)(9,10,15,20,14,21,16,19)(11,17,30,33,13,29,23,37)(12,24,27,18,28,38,31,36)(22,39,25,35,26,34,40,32);(1,3,4,7)(2,5,6,8)(9,11,14,13)(10,17,21,29)(12,25,28,40)(15,23,16,30)(18,34,36,39)(19,33,20,37)(22,31,26,27)(24,32,38,35);(9,12,15,27,14,28,16,31)(10,18,19,24,21,36,20,38)(11,22,23,40,13,26,30,25)(17,32,33,39,29,35,37,34);(2,6)(5,8)(9,13,14,11)(10,17,21,29)(12,26,28,22)(15,23,16,30)(18,35,36,32)(19,37,20,33)(24,39,38,34)(25,27,40,31)

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=f,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,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, 377937);