#148885

order	512
nilpotency class	4
isoclinism	isoclinic class of #140408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1949
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11231
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

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,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, 56081);