#135745

order	512
nilpotency class	4
isoclinism	isoclinic class of #13459
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#498
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8809
prime radical	C₂³	#430
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=e,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,e^b=ei,f^b=fi,d^c=di

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, 42941);