#148887

order	512
nilpotency class	4
isoclinism	isoclinic class of #13522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2088
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11231
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

467 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 56083);