#143941

order	512
nilpotency class	4
isoclinism	isoclinic class of #143939
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

425 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,d^c=dg,e^c=eh,f^c=fgi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51137);