#137472

order	512
nilpotency class	4
isoclinism	isoclinic class of #137383
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

295 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2,5,9,6,3,7,11)(4,10,13,16,12,8,14,15)(17,18,22,28,47,50,45,27,24,30,44,61,23,29,21,43)(19,31,34,25,48,52,33,62,36,65,72,54,35,59,46,53)(20,37,40,58,49,60,38,57,42,55,75,26,41,56,39,63)(32,51,68,64,73,80,67,76,70,79,71,74,69,78,66,77);(1,3,5,11,6,2,7,9)(4,10,14,15,12,8,13,16)(17,19,23,48,24,36,47,35)(18,25,50,53,30,54,29,62)(20,32,49,69,42,70,41,73)(21,34,44,33,45,72,22,46)(26,51,57,80,58,79,63,78)(27,59,61,65,43,52,28,31)(37,74,56,77,55,64,60,76)(38,71,75,67,39,68,40,66);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,16)(11,15)(17,20,24,42)(18,26,30,58)(19,32,36,70)(21,40,45,75)(22,39,44,38)(23,49,47,41)(25,51,54,79)(27,60,43,56)(28,55,61,37)(29,63,50,57)(31,64,65,74)(33,71,46,68)(34,66,72,67)(35,73,48,69)(52,77,59,76)(53,80,62,78)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44668);