#6347384

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347363
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	#62823	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2662 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=i,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,e^b=eh,f^b=fh,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, 6254580);