#6524724

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
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	#62836	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2900 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,28,32,44)(20,33)(21,37)(22,26)(23,27)(24,31,40,30)(25,35)(29,45,34,48)(36,42)(38,43)(39,47,41,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,41)(26,44)(27,40)(28,37)(29,35)(33,39)(36,46)(38,47)(42,45)(43,48);(2,7)(3,9)(4,11)(12,16)(17,20)(18,25)(19,29)(21,36)(22,38)(23,35)(24,39)(26,42)(27,33)(28,45)(30,46)(31,47)(32,34)(37,43)(40,41)(44,48);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,12,13,16)(20,35)(25,33)(29,34)(36,38)(39,41)(42,43)(45,48)(46,47);(3,9)(6,13)(10,15)(12,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,36,35,38)(24,28,40,44)(25,42,33,43)(29,46,34,47)(39,45,41,48)

Polycyclic presentation:

2^9|a^2=gh,d^2=hi,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, 6431920);