#148803

order	512
nilpotency class	4
isoclinism	isoclinic class of #140362
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1941
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11220
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

475 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)(3,14)(4,20,25,60)(5,26)(6,30)(7,13)(8,16)(9,23,41,24)(10,29)(11,28)(12,46)(15,50,54,63)(17,35)(18,38)(19,37)(21,49,53,62)(22,45,27,48)(31,47)(32,42,51,40)(33,43)(34,61,59,57)(36,58)(39,52,64,55)(44,56);(1,3,5,16,7,19,29,18)(2,8,10,14,13,38,26,37)(4,21,22,61,25,53,27,57)(6,31,28,17,33,56,46,58)(9,39,40,63,41,64,42,50)(11,44,30,35,12,47,43,36)(15,51,52,24,54,32,55,23)(20,59,48,49,60,34,45,62);(1,4,7,25)(2,9,13,41)(3,15)(5,27,29,22)(6,32,33,51)(8,34)(10,42,26,40)(11,45,12,48)(14,49)(16,55)(17,57)(18,52)(19,54)(20,30,60,43)(21,31)(23,28,24,46)(35,63)(36,50)(37,62)(38,59)(39,47)(44,64)(53,56)(58,61)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fh,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, 55999);