#6529580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500157
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	#62825	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2734 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,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, 6436776);