#493640

order	512
nilpotency class	4
isoclinism	isoclinic class of #493638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20609
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#63063	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2174 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)(3,13)(4,9)(5,22,24,10)(6,25,26,11)(7,12)(8,16,29,45)(14,39)(15,42,44,37)(17,38)(18,36)(19,48,50,31)(20,35,51,32)(21,33)(23,34)(27,41,30,47)(28,43)(40,59,46,60)(49,57)(52,62)(53,54)(55,61)(56,63)(58,64);(1,3,7,17)(2,8,12,29)(4,18,21,14)(5,15,24,44)(6,16,26,45)(9,30,33,27)(10,28,22,52)(11,13,25,38)(19,46,50,40)(20,47,51,41)(23,43,54,62)(31,56,48,55)(32,36,35,39)(34,37,53,42)(49,63,64,61)(57,59,58,60);(1,4,7,21)(2,9,12,33)(3,14,17,18)(5,19,24,50)(6,20,26,51)(8,27,29,30)(10,31,22,48)(11,35,25,32)(13,36,38,39)(15,40,44,46)(16,41,45,47)(23,49,54,64)(28,55,52,56)(34,58,53,57)(37,59,42,60)(43,61,62,63);(1,5)(2,10)(3,15)(4,19)(6,23)(7,24)(8,28)(9,31)(11,34)(12,22)(13,37)(14,40)(16,43)(17,44)(18,46)(20,49)(21,50)(25,53)(26,54)(27,55)(29,52)(30,56)(32,57)(33,48)(35,58)(36,59)(38,42)(39,60)(41,61)(45,62)(47,63)(51,64)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400836);