#6354910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354835
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60900	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1554 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)(17,18,23,28)(19,29,33,39)(20,34,36,41)(21,26,22,27)(24,31,40,32)(25,37,42,35)(30,44,46,38)(43,45,48,47);(1,3,4,11,6,13,5,12)(2,7,14,16,9,15,8,10)(17,19,22,31,23,33,21,32)(18,24,26,39,28,40,27,29)(20,30,37,45,36,46,35,47)(25,38,41,48,42,44,34,43);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,47)(33,46)(39,48)(40,44);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,29,40,39)(25,41,42,34)(30,45,46,47)(38,43,44,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(20,36)(25,42)(30,46)(34,41)(35,37)(38,44)(43,48)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262106);