#148630

order	512
nilpotency class	4
isoclinism	isoclinic class of #13502
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11199
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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ehi,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, 55826);