#6491467

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62851	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2911 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)(9,10,15,27,14,17,13,19)(11,20,26,16,23,33,22,29)(12,18,28,38,24,31,25,32)(21,34,37,30,35,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,26)(14,23)(15,22)(17,29)(18,30)(19,33)(20,27)(24,35)(25,37)(28,36)(31,39)(32,40)(34,38);(2,5)(6,8)(10,17)(13,15)(16,29)(18,31)(22,26)(25,28)(30,39)(36,37);(2,5)(6,8)(9,12,14,24)(10,18,17,31)(11,21,23,35)(13,25,15,28)(16,30,29,39)(19,32,27,38)(20,34,33,40)(22,36,26,37);(1,3,4,7)(2,6,5,8)(12,24)(18,31)(21,35)(25,28)(30,39)(32,38)(34,40)(36,37)

Polycyclic presentation:

2^9|a^2=fi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=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, 6398663);