#6347186

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347145
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2751 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,d^b=dh,f^b=fh,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, 6254382);