#127408

order	512
nilpotency class	3
isoclinism	isoclinic class of #127312
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×H(3,ℤ/4)
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	#405	C₂³
socle	C₂	C₂×#1290
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

319 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ghi,e^2=g,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=ci,e^b=eh,e^d=ei

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, 34604);