#6530956

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
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	#62838	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(17,18)(19,28,32,44)(20,25)(21,36)(22,26)(23,27)(24,31,39,30)(29,46,33,45)(34,41)(35,43)(37,42)(38,47,40,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,31)(22,30)(23,32)(25,40)(26,44)(27,39)(28,36)(29,34)(35,47)(37,48)(38,41)(42,45)(43,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,23,34)(18,25,27,41)(19,29,32,33)(21,35,22,37)(24,38,39,40)(26,42,36,43)(28,45,44,46)(30,47,31,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(20,34)(25,41)(29,33)(35,37)(38,40)(42,43)(45,46)(47,48);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,35,34,37)(24,28,39,44)(25,42,41,43)(29,47,33,48)(38,45,40,46)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,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, 6438152);