#498436

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

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,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, 405632);