#6519825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
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	#60806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6427021);