#148777

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2067
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11219
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

467 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)(3,14)(4,20,25,58)(5,27)(6,30)(7,13)(8,16)(9,23,41,24)(10,29)(11,28)(12,46)(15,50,53,62)(17,35)(18,38)(19,37)(21,49,52,63)(22,45,26,48)(31,47)(32,42,60,40)(33,43)(34,54,61,59)(36,56)(39,57,64,51)(44,55);(1,3,5,16,7,19,29,18)(2,8,10,14,13,38,27,37)(4,21,22,59,25,52,26,54)(6,31,28,56,33,55,46,17)(9,39,40,50,41,64,42,62)(11,44,30,36,12,47,43,35)(15,32,51,23,53,60,57,24)(20,34,48,49,58,61,45,63);(1,4,5,22,7,25,29,26)(2,9,10,40,13,41,27,42)(3,15,16,51,19,53,18,57)(6,32,28,23,33,60,46,24)(8,34,14,49,38,61,37,63)(11,45,30,20,12,48,43,58)(17,54,31,21,56,59,55,52)(35,50,44,64,36,62,47,39)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,e^c=eh,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, 55973);