#510457

order	512
nilpotency class	4
isoclinism	isoclinic class of #510456
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31613
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3060	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

389 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417653);