#502024

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	#33610	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,27)(13,32)(14,21)(15,34)(16,22)(17,20)(19,23)(24,28)(25,30)(26,38)(29,36)(31,40)(33,39)(35,37);(1,3,4,7)(2,5,6,8)(9,10,17,32,15,21,33,24,16,22,28,39,14,34,13,20)(11,23,27,12,19,18,36,30,26,37,40,31,35,38,25,29);(2,6)(5,8)(9,11)(10,18)(12,20)(13,25)(14,35)(15,19)(16,26)(17,27)(21,37)(22,38)(23,34)(24,31)(28,40)(29,39)(30,32)(33,36);(9,12,28,38,15,30,13,23,16,31,17,18,14,29,33,37)(10,19,39,25,21,26,20,27,22,35,32,36,34,11,24,40)

Polycyclic presentation:

2^9|b^2=efh,d^2=ehi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,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, 409220);