#148968

order	512
nilpotency class	4
isoclinism	isoclinic class of #13502
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#11257
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

435 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=i,e^2=i,f^2=h,g^2=i,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=fh,h^b=hi,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, 56164);