#502019

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
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

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

Polycyclic presentation:

2^9|b^2=efh,d^2=eh,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,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409215);