#6447139

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354335);