#6491453

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
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	#62788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2913 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)(9,10,15,29,14,18,13,19)(11,20,27,16,23,33,22,31)(12,24,28,32,26,17,25,34)(21,35,38,39,37,40,36,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,27)(14,23)(15,22)(17,30)(18,31)(19,33)(20,29)(24,39)(25,38)(26,37)(28,36)(32,35)(34,40);(2,5)(6,8)(9,12,14,26)(10,17,18,24)(11,21,23,37)(13,28,15,25)(16,30,31,39)(19,34,29,32)(20,35,33,40)(22,38,27,36);(2,5)(6,8)(10,18)(16,31)(17,24)(19,29)(20,33)(30,39)(32,34)(35,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 6398649);