#6532518

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

2802 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=h,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, 6439714);