#139801

order	512
nilpotency class	4
isoclinism	isoclinic class of #139724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

634 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46997);