#148156

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11066
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

675 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=h,g^2=i,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=fh,h^b=hi,d^c=di,e^c=eh,f^c=fhi,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, 55352);