#498429

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

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	#3035	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

560 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)(9,10,14,20,15,19,33,38,16,21,34,27,13,30,32,39)(11,17,23,35,24,31,37,40,25,12,26,29,22,28,18,36);(1,2,5,6)(3,7,4,8)(9,11,14,23,15,24,33,37,16,25,34,26,13,22,32,18)(10,17,20,35,19,31,38,40,21,12,27,29,30,28,39,36);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,20)(13,31)(14,29)(15,28)(16,17)(19,23)(21,37)(22,39)(24,38)(25,27)(26,30)(32,40)(33,36)(34,35);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,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, 405625);