#127406

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ghi,e^2=g,f^2=gh,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, 34602);