#500116

order	512
nilpotency class	4
isoclinism	isoclinic class of #500115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24005
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60462	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,14)(3,9,18,34)(4,19,21,53)(5,25,27,42)(6,13,7,12)(10,35,37,22)(11,28,41,29)(15,44,45,64)(16,47,49,63)(17,43,30,33)(20,55,51,52)(23,54,24,40)(26,39,57,38)(31,59,60,46)(32,58,61,50)(36,56,62,48);(1,3)(2,9)(4,20)(5,16)(6,30)(7,17)(8,18)(10,36)(11,32)(12,43)(13,33)(14,34)(15,23)(19,52)(21,51)(22,56)(24,45)(25,47)(26,46)(27,49)(28,58)(29,50)(31,38)(35,48)(37,62)(39,60)(40,44)(41,61)(42,63)(53,55)(54,64)(57,59);(1,4,7,23,8,21,6,24)(2,10,13,38,14,37,12,39)(3,15,17,51,18,45,30,20)(5,26,29,35,27,57,28,22)(9,31,33,62,34,60,43,36)(11,40,25,53,41,54,42,19)(16,48,50,59,49,56,58,46)(32,55,47,64,61,52,63,44);(5,27)(11,41)(16,49)(19,53)(22,35)(25,42)(26,57)(28,29)(32,61)(40,54)(44,64)(46,59)(47,63)(48,56)(50,58)(52,55)

Polycyclic presentation:

2^9|a^2=i,c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407312);