#149061

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11292
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

445 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,f^b=fh,g^b=gh,h^b=hi,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, 56257);