#148851

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	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

483 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,g^2=hi,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, 56047);