#497124

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22261
prime radical	#23794	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

558 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,f^b=fi,d^c=dh,e^c=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224961, #72320755, C₂×#497124
maximal normal subgroup of	#72224961, #72320755, C₂×#497124
maximal quotient of	C₂×#497124

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404320);