#504717

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3155	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

523 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)(11,25)(12,13)(14,21)(15,20)(16,22)(17,24)(18,19)(23,35)(26,39)(27,40)(28,34)(29,33)(30,36)(31,32)(37,38);(1,3,5,4)(2,6,7,8)(9,11,26,31,16,28,14,19,17,29,15,27,35,12,30,38)(10,18,36,34,23,32,20,25,24,37,21,13,22,40,39,33);(1,4,5,3)(2,6,7,8)(9,12,14,31,35,29,26,38,17,28,30,27,16,11,15,19)(10,18,20,25,22,40,36,34,24,37,39,33,23,32,21,13);(2,7)(6,8)(9,13,26,40,16,33,14,18,17,34,15,32,35,25,30,37)(10,19,36,29,23,27,20,12,24,38,21,11,22,31,39,28)

Polycyclic presentation:

2^9|b^2=ehi,c^2=e,d^2=fi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411913);