#148638

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
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

625 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=dh,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fi,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, 55834);