#148915

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2054
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11232
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

459 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,f^c=fi

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, 56111);