#6436353

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^d=ei

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, 6343549);