#148657

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11200
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

627 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)(3,14,18,39)(4,20,25,54)(5,10)(6,28)(7,13)(8,16,34,19)(9,23,38,59)(11,26)(12,27)(15,45)(17,41,29,33)(21,55)(22,53,24,42)(30,37,58,36)(31,40)(32,50)(35,64)(43,57)(44,49,60,51)(46,62)(47,61)(48,52)(56,63);(1,3)(2,8)(4,21)(5,19)(6,29)(7,18)(9,35)(10,39)(11,41)(12,14)(13,34)(15,24)(16,27)(17,31)(20,52)(22,48)(23,56)(25,46)(26,49)(28,60)(30,47)(32,37)(33,43)(36,63)(38,61)(40,44)(42,55)(45,54)(50,59)(51,57)(53,62)(58,64);(1,4,7,25)(2,9)(3,15)(5,24,27,22)(6,30,31,58)(8,32,34,63)(10,37)(11,42)(12,36)(13,38)(14,35,39,61)(16,21)(17,50)(18,48)(19,46)(20,28)(23,57,59,26)(29,56)(33,52,41,45)(40,54)(43,53)(44,62,60,55)(47,51)(49,64)

Polycyclic presentation:

2^9|a^2=g,c^2=dh,d^2=i,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=eh,g^c=gi,h^c=hi

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, 55853);