#332644

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239840);