#507197

order	512
nilpotency class	4
isoclinism	isoclinic class of #502472
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	#33612	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

563 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)(12,26)(13,18)(14,21)(15,20)(16,22)(17,24)(19,35)(23,27)(28,32)(29,31)(30,39)(33,34)(36,38)(37,40);(1,3,5,4)(2,6,7,8)(9,11,26,36,27,35,15,28,17,30,14,13,16,29,34,40)(10,18,20,25,22,32,12,31,24,38,33,39,23,37,21,19);(1,4,5,3)(2,6,7,8)(9,12,14,24,27,33,34,23,17,21,26,10,16,20,15,22)(11,25,13,32,35,31,40,38,30,39,36,37,29,19,28,18);(2,7)(6,8)(9,13,34,30,16,28,14,35,17,36,15,11,27,40,26,29)(10,19,12,32,23,39,20,18,24,31,21,37,22,25,33,38)

Polycyclic presentation:

2^9|b^2=egh,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,d^b=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, 414393);