#498250

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

507 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)(3,6)(4,5)(7,8)(9,10,15,21,16,19,33,38,14,20,32,22,13,30,31,39)(11,12,26,29,23,28,18,36,24,17,34,35,25,27,37,40);(2,5)(3,7)(9,11,15,26,16,23,33,18,14,24,32,34,13,25,31,37)(10,17,21,35,19,27,38,40,20,12,22,29,30,28,39,36);(9,12)(10,18)(11,22)(13,27)(14,17)(15,29)(16,28)(19,34)(20,37)(21,24)(23,39)(25,38)(26,30)(31,40)(32,35)(33,36);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,19,20,30)(11,23,24,25)(12,27,17,28)(15,31,32,33)(18,34,37,26)(21,38,22,39)(29,40,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 405446);