#334783

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	#25741	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

757 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)(12,24)(13,29)(14,19)(15,20)(17,26)(18,22)(21,35)(23,33)(25,31)(27,36)(28,37)(30,32)(34,38)(39,40);(1,3,5,2)(4,8,6,7)(9,11)(10,16)(12,25)(13,22)(14,30)(15,23)(17,34)(18,29)(19,32)(20,33)(21,27)(24,31)(26,38)(28,39)(35,36)(37,40);(2,3)(7,8)(9,12,15,28)(10,17,20,36)(11,21,23,38)(13,27,14,26)(16,31,33,40)(18,35,32,34)(19,37,29,24)(22,39,30,25);(2,7)(3,8)(4,6)(9,11)(10,18)(12,21)(13,30)(14,22)(15,23)(16,29)(17,35)(19,33)(20,32)(24,31)(25,27)(26,39)(28,38)(34,36)(37,40)

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^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, 241979);