#334809

order	512
nilpotency class	3
isoclinism	isoclinic class of #334695
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

758 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242005);