#149056

order	512
nilpotency class	4
isoclinism	isoclinic class of #140203
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1934
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11292
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

445 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=i,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,f^b=fhi,g^b=ghi,h^b=hi,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, 56252);