#149114

order	512
nilpotency class	4
isoclinism	isoclinic class of #13497
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2075
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

409 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:

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

Polycyclic presentation:

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

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, 56310);