#496010

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ 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)(9,10,15,21,14,19,29,22)(11,23,26,34,12,16,27,33)(13,18,31,20,30,37,40,38)(17,28,36,24,35,39,32,25);(1,3,5,4)(2,6,7,8)(9,11,14,12)(10,16,19,23)(13,25,30,24)(15,26,29,27)(17,18,35,37)(20,32,38,36)(21,33,22,34)(28,40,39,31);(9,12,14,11)(10,17,19,35)(13,28,30,39)(15,26,29,27)(16,18,23,37)(20,33,38,34)(21,32,22,36)(24,31,25,40);(1,4,5,3)(2,6,7,8)(9,13,29,40,14,30,15,31)(10,18,22,38,19,37,21,20)(11,24,27,28,12,25,26,39)(16,17,34,32,23,35,33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403206);