#139097

order	512
nilpotency class	4
isoclinism	isoclinic class of #139074
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

301 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,22,29,47,50,45,27,23,30,46,61,24,31,21,28)(19,32,36,25,48,52,34,62,37,65,72,53,38,59,35,54)(20,39,41,57,49,60,40,58,42,55,44,26,43,56,74,63)(33,51,68,76,73,79,67,75,69,77,71,64,70,78,66,80);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19,24,48,23,37,47,38)(18,25,50,54,30,53,31,62)(20,33,49,70,42,69,43,73)(21,36,46,34,45,72,22,35)(26,51,58,79,57,77,63,78)(27,59,61,65,28,52,29,32)(39,64,56,80,55,76,60,75)(40,71,44,67,74,68,41,66);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(17,20,23,42)(18,26,30,57)(19,33,37,69)(21,44,45,41)(22,40,46,74)(24,49,47,43)(25,51,53,77)(27,56,28,60)(29,39,61,55)(31,63,50,58)(32,64,65,76)(34,68,35,71)(36,67,72,66)(38,73,48,70)(52,75,59,80)(54,79,62,78)

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=i,d^2=ghi,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=ehi,f^c=fgh,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, 46293);