#509299

order	512
nilpotency class	4
isoclinism	isoclinic class of #509298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30567
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28181	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2930 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

519 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416495);