#146293

order	512
nilpotency class	4
isoclinism	isoclinic class of #146293
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10691
commutator, Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

215 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:

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

Polycyclic presentation:

2^9|a^2=defg,b^2=de,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,f^a=fh,g^a=gi,c^b=cf,d^b=dh,e^b=eh,f^b=fgi,e^c=eg,f^c=fhi,h^c=hi,f^d=fi,f^e=fi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53489);