#139115

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	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1746 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 346 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=i,d^2=ghi,e^2=h,f^2=ghi,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, 46311);