#148796

order	512
nilpotency class	4
isoclinism	isoclinic class of #13400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11220
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

469 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=hi,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=ehi,f^c=fh,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, 55992);