#139161

order	512
nilpotency class	4
isoclinism	isoclinic class of #139160
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

301 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46357);