#148792

order	512
nilpotency class	4
isoclinism	isoclinic class of #13460
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2058
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11220
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

473 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fh,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, 55988);