#501859

order	512
nilpotency class	4
isoclinism	isoclinic class of #501858
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24897
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

626 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:

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

Polycyclic presentation:

2^9|b^2=fh,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,e^b=eh,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409055);