#509179

order	512
nilpotency class	4
isoclinism	isoclinic class of #509179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30444
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

774 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:

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

Polycyclic presentation:

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

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, 416375);