#139116

order	512
nilpotency class	4
isoclinism	isoclinic class of #139080
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 ◁ #1746 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 342 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=ehi,f^c=fghi,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, 46312);