#332772

order	512
nilpotency class	3
isoclinism	isoclinic class of #332711
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

603 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)(12,18,27,34)(13,16,14,19)(17,28,32,22)(23,37,38,35)(25,30,26,33)(31,36,40,39);(1,2,5,6)(3,7,8,4)(10,16)(11,22)(12,25)(13,14)(19,20)(21,29)(23,38)(24,28)(26,27)(30,33)(31,35)(37,40);(1,3,5,8)(2,7,6,4)(9,11,15,24)(10,17,20,32)(12,23,27,38)(13,22,14,28)(16,29,19,21)(18,31,34,40)(25,36,26,39)(30,37,33,35);(1,4)(2,3)(5,7)(6,8)(9,12,15,27)(10,18,20,34)(11,23,24,38)(13,25,14,26)(16,30,19,33)(17,31,32,40)(21,35,29,37)(22,36,28,39)

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=dei,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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239968);