#6487820

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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)(4,7)(5,8)(10,16)(11,21)(13,14)(18,31)(19,20)(23,35)(24,25)(26,34)(27,29)(30,33)(37,38)(39,40);(3,4)(6,7)(9,10,13,19,15,20,14,16)(11,22,21,30,25,17,24,33)(12,26,29,31,28,18,27,34)(23,32,35,40,38,36,37,39);(1,3,5,4)(2,6,8,7)(9,11,15,25)(10,17,20,22)(12,23,28,38)(13,21,14,24)(16,30,19,33)(18,32,26,36)(27,37,29,35)(31,39,34,40);(1,4,5,3)(2,7,8,6)(9,12,15,28)(10,18,20,26)(11,23,25,38)(13,29,14,27)(16,31,19,34)(17,32,22,36)(21,35,24,37)(30,39,33,40);(2,8)(6,7)(9,13,15,14)(10,19,20,16)(11,21,25,24)(12,27,28,29)(17,33,22,30)(18,31,26,34)(23,37,38,35)(32,39,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6395016);