#149347

order	512
nilpotency class	4
isoclinism	isoclinic class of #135291
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1825
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11358
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

431 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)(3,16,21,39)(4,23,28,58)(5,30)(6,34)(7,13)(8,15)(9,18,41,19)(10,26,44,59)(11,33)(12,37)(14,25)(17,38)(20,47,22,45)(24,42)(27,48,29,46)(31,49,35,40)(32,50,36,43)(51,60)(52,64)(53,63)(54,57)(55,61)(56,62);(1,3)(2,9)(4,24,28,56)(5,31)(6,35)(7,20)(8,21)(10,42,44,62)(11,45)(12,47)(13,40)(14,49)(15,41)(16,34)(17,27,53,29)(18,37)(19,33)(22,25)(23,57,58,61)(26,52,59,60)(30,39)(32,55,36,54)(38,43,63,50)(46,51,48,64);(1,4,25,29,8,28,7,27)(2,10,13,43,15,44,14,50)(3,17,20,24,21,53,22,56)(5,32,37,26,6,36,33,59)(9,38,49,62,41,63,40,42)(11,46,34,23,12,48,30,58)(16,51,45,61,39,64,47,57)(18,54,31,52,19,55,35,60)

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,g^b=gi,h^b=hi,d^c=dhi,e^c=eh,g^c=gi,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, 56543);