#148461

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#394
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11147
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

637 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55657);