#143293

order	512
nilpotency class	4
isoclinism	isoclinic class of #143212
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

359 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,7)(4,8)(9,10,15,20)(11,21,24,53)(12,26,29,17)(13,31,36,51)(14,37,34,19)(16,39,41,23)(18,46,49,35)(22,52,27,59)(25,57,38,47)(28,61,33,66)(30,45,63,58)(32,50,65,42)(40,54,43,56)(44,64,48,68)(55,72,60,71)(62,70,67,69);(1,2,5,3)(4,7,8,6)(9,11,15,24)(10,16,20,41)(12,27,29,22)(13,32,36,65)(14,38,34,25)(17,43,26,40)(18,47,49,57)(19,50,37,42)(21,51,53,31)(23,46,39,35)(28,62,33,67)(30,55,63,60)(44,71,48,72)(45,69,58,70)(52,61,59,66)(54,64,56,68);(1,3,5,2)(4,6,8,7)(9,12,14,30,15,29,34,63)(10,17,19,45,20,26,37,58)(11,22,25,55,24,27,38,60)(13,33,35,68,36,28,46,64)(16,40,42,69,41,43,50,70)(18,48,31,61,49,44,51,66)(21,52,47,71,53,59,57,72)(23,56,32,62,39,54,65,67)

Polycyclic presentation:

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

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, 50489);