#495972

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
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

536 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)(3,8)(4,6)(5,7)(9,10,14,19,15,18,32,37,16,20,33,25,13,29,31,39)(11,21,23,27,22,17,34,28,24,35,38,26,30,36,40,12);(1,3,5,4)(2,6,7,8);(9,11)(10,17)(12,25)(13,30)(14,23)(15,22)(16,24)(18,35)(19,28)(20,36)(21,29)(26,37)(27,39)(31,40)(32,34)(33,38);(1,4,5,3)(2,6,7,8)(9,12,14,21,15,27,32,17,16,28,33,35,13,26,31,36)(10,11,19,23,18,22,37,34,20,24,25,38,29,30,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403168);