#501865

order	512
nilpotency class	4
isoclinism	isoclinic class of #501864
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24909
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2193	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

621 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,f^c=fi,e^d=eh,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, 409061);