#148539

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

643 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)(3,15,19,45)(4,21)(5,12)(6,30)(7,33)(8,14)(9,17,37,54)(10,24)(11,28)(13,32)(16,35,52,62)(18,48,55,38)(20,44,34,36)(22,59,50,64)(23,42)(25,58)(26,41)(27,46)(29,47)(31,43)(39,51,49,56)(40,61)(53,57,60,63);(1,3)(2,9)(4,22)(5,18)(6,17)(7,34)(8,19)(10,39)(11,44)(12,45)(13,48)(14,37)(15,47)(16,25)(20,32)(21,57)(23,52)(24,53)(26,51)(27,50)(28,54)(29,55)(30,36)(31,60)(33,38)(35,41)(40,62)(42,64)(43,49)(46,63)(56,61)(58,59);(1,4,5,23,7,26,28,24,8,27,29,25,32,61,6,31)(2,10,11,40,13,42,12,46,14,43,30,41,33,58,47,21)(3,16,17,50,20,53,55,51,19,52,54,22,34,60,18,56)(9,35,15,49,38,63,36,64,37,62,45,39,48,57,44,59)

Polycyclic presentation:

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

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, 55735);