#496034

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21635
prime radical	#575	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=g,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224961, C₂×#496034
maximal normal subgroup of	#72224961, C₂×#496034
maximal quotient of	C₂×#496034

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403230);