#148884

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	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

459 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)(3,15,20,45)(4,21)(5,27)(6,30)(7,13)(8,33)(9,14)(10,17,36,18)(11,24)(12,29)(16,38,50,62)(19,44,31,46)(22,58,51,34)(23,41)(25,55)(26,57)(28,42)(32,56)(35,52,37,53)(39,61)(40,47)(43,48,63,49)(54,59,64,60);(1,3,5,17,8,19,29,52,9,20,6,18,7,31,28,53)(2,10,12,15,13,37,27,46,14,36,42,45,33,35,30,44)(4,22,23,59,25,16,47,48,26,51,24,60,32,50,61,49)(11,38,21,54,40,58,56,63,41,62,57,64,39,34,55,43);(1,4,5,23,8,25,29,47,9,26,6,24,7,32,28,61)(2,11,12,21,13,40,27,56,14,41,42,57,33,39,30,55)(3,16,17,48,19,51,52,60,20,50,18,49,31,22,53,59)(10,34,15,43,37,38,46,54,36,58,45,63,35,62,44,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^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, 56080);