#510328

order	512
nilpotency class	4
isoclinism	isoclinic class of #510327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31495
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

442 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)(6,28,31,42,32,13,27,43)(16,46,19,52,29,37,51,34)(17,48,22,55,49,35,56,39)(25,58,59,44,30,54,57,40)(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)(6,29,27,51,32,16,31,19)(11,39,21,48,38,55,41,35)(13,34,43,37,28,52,42,46)(25,53,59,47,30,50,57,60)(40,63,54,61,44,62,58,64);(2,10)(3,16)(5,7)(6,27)(9,34)(11,38)(13,43)(15,37)(17,47)(18,51)(19,33)(20,29)(22,50)(23,26)(25,57)(28,42)(30,59)(31,32)(35,61)(36,52)(39,63)(40,58)(44,54)(45,46)(48,64)(49,60)(53,56)(55,62);(1,4,8,24)(2,11,10,38)(3,17)(5,23,7,26)(6,30,32,25)(9,35)(12,21,14,41)(13,44,28,40)(15,39)(16,47)(18,22)(19,53)(20,49)(27,59,31,57)(29,60)(33,56)(34,61)(36,48)(37,63)(42,58,43,54)(45,55)(46,62)(50,51)(52,64)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,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=ghi,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, 417524);