#493485

order	512
nilpotency class	4
isoclinism	isoclinic class of #493483
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20585
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#62887	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #3485 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2166 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)(3,13,17,37)(4,9)(5,10)(6,24,26,11)(7,12)(8,16)(14,38,18,44)(15,39,43,59)(19,31)(20,49,50,32)(21,33)(22,51,53,35)(23,36)(25,29)(27,41)(28,42)(30,46)(34,48)(40,60,45,61)(47,58,64,57)(52,56)(54,62)(55,63);(1,3,7,17)(2,8,12,29)(4,18,21,14)(5,15,23,43)(6,25,26,16)(9,30,33,27)(10,28,36,56)(11,37,24,13)(19,45,48,40)(20,41,50,46)(22,52,53,42)(31,55,34,54)(32,38,49,44)(35,59,51,39)(47,62,64,63)(57,60,58,61);(1,4,7,21)(2,9,12,33)(3,14,17,18)(5,19,23,48)(6,20,26,50)(8,27,29,30)(10,34,36,31)(11,32,24,49)(13,38,37,44)(15,40,43,45)(16,41,25,46)(22,47,53,64)(28,55,56,54)(35,58,51,57)(39,61,59,60)(42,62,52,63);(1,5)(2,10)(3,15)(4,19)(6,22)(7,23)(8,28)(9,31)(11,35)(12,36)(13,39)(14,40)(16,42)(17,43)(18,45)(20,47)(21,48)(24,51)(25,52)(26,53)(27,54)(29,56)(30,55)(32,57)(33,34)(37,59)(38,60)(41,62)(44,61)(46,63)(49,58)(50,64)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,e^b=ei,g^b=ghi,d^c=dh,f^d=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, 400681);