#148853

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:

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

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,e^a=ei,f^a=fi,c^b=cf,e^b=ei,e^c=eh,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, 56049);