#498393

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 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)(7,8)(9,10,16,22,15,19,33,38,14,20,32,23,13,21,31,39)(11,12,26,28,24,29,37,40,25,17,34,35,30,27,18,36);(1,2)(3,7)(4,8)(5,6)(9,11,16,26,15,24,33,37,14,25,32,34,13,30,31,18)(10,17,22,35,19,27,38,36,20,12,23,28,21,29,39,40);(1,3,5,4)(2,7,6,8)(9,12,14,17)(10,18,20,37)(11,23,25,22)(13,27,15,29)(16,28,32,35)(19,26,21,34)(24,39,30,38)(31,36,33,40);(1,4,5,3)(2,7,6,8)(9,13,14,15)(10,19,20,21)(11,24,25,30)(12,27,17,29)(16,31,32,33)(18,26,37,34)(22,38,23,39)(28,36,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405589);