#139215

order	512
nilpotency class	4
isoclinism	isoclinic class of #139156
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	#121	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

313 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46411);