#148813

order	512
nilpotency class	4
isoclinism	isoclinic class of #140094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1919
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11220
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 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)(3,15)(4,22,27,47)(5,28)(6,13)(7,35)(8,14)(9,17)(10,25,44,58)(11,31)(12,30)(16,41,52,63)(18,37)(19,38)(20,40)(21,39)(23,36,57,62)(24,43,29,56)(26,42,33,45)(32,59)(34,48)(46,54)(49,51,60,55)(50,64,53,61);(1,3,5,17,8,21,31,20)(2,9,11,15,14,40,28,39)(4,23,24,55,27,57,29,51)(6,32,30,54,34,19,7,18)(10,41,42,61,44,63,45,64)(12,38,13,46,35,59,48,37)(16,25,50,26,52,58,53,33)(22,49,56,36,47,60,43,62);(1,4,8,27)(2,10,14,44)(3,16)(5,29,31,24)(6,33,34,26)(7,25,30,58)(9,36)(11,45,28,42)(12,43,35,56)(13,47,48,22)(15,49)(17,53)(18,23)(19,55)(20,50)(21,52)(32,51)(37,63)(38,61)(39,60)(40,62)(41,46)(54,57)(59,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,d^c=di,e^c=ehi,f^c=fhi,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, 56009);