#6606654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574100
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	#61256	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 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:

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

Polycyclic presentation:

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

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, 6513850);