#148850

order	512
nilpotency class	4
isoclinism	isoclinic class of #141878
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1988
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11227
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

475 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=ghi,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, 56046);