#149415

order	512
nilpotency class	4
isoclinism	isoclinic class of #140776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11379
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

431 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=deh,d^2=hi,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,h^b=hi,d^c=dhi,e^c=ehi,f^c=fh,g^c=gh,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, 56611);