#149073

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11293
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

455 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=gh,h^b=hi,e^c=ei,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, 56269);