#149141

order	512
nilpotency class	4
isoclinism	isoclinic class of #140140
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1925
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

413 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,f^c=fi

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, 56337);