#355561

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13785
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1281 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,h^a=hi,c^b=ce,f^b=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262757);