#332854

order	512
nilpotency class	3
isoclinism	isoclinic class of #332839
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

750 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,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, 240050);