#148960

order	512
nilpotency class	4
isoclinism	isoclinic class of #13504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11257
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

439 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fh,g^b=gi,h^b=hi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56156);