#6436612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432876
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	#62686	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2913 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)(9,12)(10,13)(11,14)(15,16)(17,18,23,27)(19,28,31,38)(20,25,33,41)(21,26,22,34)(24,35,39,30)(29,44,47,43)(32,40,36,42)(37,45,48,46);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19)(18,24)(20,29)(21,30)(22,35)(23,31)(25,37)(26,38)(27,39)(28,34)(32,45)(33,47)(36,46)(40,44)(41,48)(42,43);(3,9)(6,12)(10,15)(13,16)(17,20)(18,25)(19,29)(21,32)(22,36)(23,33)(24,37)(26,40)(27,41)(28,43)(30,45)(31,47)(34,42)(35,46)(38,44)(39,48);(2,7)(6,12)(8,14)(13,16)(17,21,23,22)(18,26,27,34)(19,30,31,35)(20,32,33,36)(24,38,39,28)(25,40,41,42)(29,45,47,46)(37,44,48,43);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,13,12,16)(18,27)(24,39)(25,41)(26,34)(28,38)(37,48)(40,42)(43,44)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,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, 6343808);