#148870

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	#11228
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

481 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,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,f^b=fi,d^c=di,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, 56066);