#6572284

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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	#62961	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2972 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:

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

Polycyclic presentation:

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

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, 6479480);