#148964

order	512
nilpotency class	4
isoclinism	isoclinic class of #13512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2085
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:

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

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,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,f^c=fh,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, 56160);