#304209

order	512
nilpotency class	3
isoclinism	isoclinic class of #267659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

844 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=ef,d^2=ehi,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211405);