#501070

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32933	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,h^c=hi

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, 408266);