#510118

order	512
nilpotency class	4
isoclinism	isoclinic class of #510115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31314
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

554 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=efh,c^2=i,d^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=ghi,h^c=hi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417314);