#505232

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25805	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=ei,d^2=f,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, 412428);