#149081

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	#11293
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

453 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:

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

Polycyclic presentation:

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

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, 56277);