#6494250

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473167
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	#60919	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1702 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,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, 6401446);