#405743

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1121 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)(3,8)(4,6)(5,7)(9,10,15,20)(11,16,23,32)(12,24,28,34)(13,19,14,18)(17,27,35,26)(21,33,37,39)(22,36,29,31)(25,30,38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,25,28,38)(13,29)(14,22)(15,23)(17,33,35,39)(18,36)(19,31)(20,32)(21,27,37,26)(24,30,34,40);(9,12)(10,17)(11,21)(13,27)(14,26)(15,28)(16,30)(18,34)(19,24)(20,35)(22,38)(23,37)(25,29)(31,39)(32,40)(33,36);(1,4,5,3)(2,6,7,8)(9,11)(10,16)(12,21)(13,29)(14,22)(15,23)(17,30)(18,36)(19,31)(20,32)(24,39)(25,27)(26,38)(28,37)(33,34)(35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi,f^d=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, 312939);