#139093

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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 307 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) = 128

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=i,d^2=ghi,e^2=h,f^2=h,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, 46289);