#509610

order	512
nilpotency class	4
isoclinism	isoclinic class of #509608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30846
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

623 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)(3,17)(4,22)(5,29)(6,33)(7,15)(8,14)(9,16)(10,19)(11,26)(12,21)(13,31)(18,50)(20,45)(23,49)(24,40)(25,59)(27,43)(28,48)(30,47)(32,38)(34,39)(35,44)(36,46)(37,55)(41,52)(42,56)(51,62)(53,63)(54,57)(58,61)(60,64);(1,3)(2,10)(4,23)(5,30)(6,34)(7,20)(8,36)(9,21)(11,40)(12,44)(13,46)(14,38)(15,39)(16,29)(17,33)(18,27)(19,31)(22,37)(24,56)(25,51)(26,53)(28,52)(32,35)(41,42)(43,62)(45,47)(48,61)(49,59)(50,64)(54,63)(55,57)(58,60);(1,4,9,28)(2,11,16,43)(3,18,21,53)(5,24,19,51)(6,25,35,56)(7,27,8,26)(10,37,29,61)(12,41,17,49)(13,42,47,59)(14,48,15,22)(20,52,36,23)(30,60,31,57)(32,55,34,58)(33,54,44,64)(38,40,39,62)(45,50,46,63);(1,5,31,21,8,32,35,36,9,19,30,3,7,34,6,20)(2,12,33,29,15,45,47,39,16,17,44,10,14,46,13,38)(4,24,57,53,26,55,56,23,28,51,60,18,27,58,25,52)(11,41,54,61,22,50,59,62,43,49,64,37,48,63,42,40)

Polycyclic presentation:

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

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, 416806);