#291004

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

831 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)(9,10,15,19)(11,16,17,23)(12,21,24,20)(13,18,14,22)(25,26,28,31)(27,35,37,40)(29,34,30,33)(32,36,39,38);(1,3,4,8)(2,5,6,7)(9,11,15,17)(10,16,19,23)(12,22,24,18)(13,20,14,21)(26,31)(27,36)(29,30)(32,35)(37,38)(39,40);(9,12,13,23,15,24,14,16)(10,17,22,20,19,11,18,21)(25,27,30,36,28,37,29,38)(26,32,34,40,31,39,33,35);(2,6)(5,7)(9,13,15,14)(10,18,19,22)(11,20,17,21)(12,23,24,16)(26,31)(32,39)(33,34)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fh,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, 198200);