#291008

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

827 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198204);