#501863

order	512
nilpotency class	4
isoclinism	isoclinic class of #501862
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	#33540	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

622 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)(6,31)(7,15)(8,14)(9,16)(10,19)(11,26)(12,30)(13,34)(18,49)(20,43)(21,48)(23,55)(24,57)(25,59)(27,41)(28,46)(32,37)(33,45)(35,44)(36,60)(38,50)(39,56)(40,63)(42,52)(47,51)(53,58)(54,61)(62,64);(1,3,7,20,9,5,8,21)(2,10,14,37,16,12,15,42)(4,23,26,18,28,51,27,24)(6,32,33,19,35,52,34,30)(11,38,22,39,41,53,46,36)(13,43,44,17,45,48,31,29)(25,50,61,60,63,58,62,56)(40,55,54,57,59,47,64,49);(1,4)(2,11)(3,18)(5,24)(6,25)(7,27)(8,26)(9,28)(10,36)(12,39)(13,40)(14,46)(15,22)(16,41)(17,47)(19,50)(20,23)(21,51)(29,55)(30,58)(31,54)(32,60)(33,62)(34,61)(35,63)(37,53)(38,42)(43,49)(44,64)(45,59)(48,57)(52,56);(1,5,9,3)(2,12,16,10)(4,24,28,18)(6,32,35,52)(7,21,8,20)(11,39,41,36)(13,43,45,48)(14,42,15,37)(17,31,29,44)(19,34,30,33)(22,53,46,38)(23,27,51,26)(25,60,63,56)(40,49,59,57)(47,54,55,64)(50,61,58,62)

Polycyclic presentation:

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

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, 409059);