#143251

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	C₂²×D₈	D₄
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

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

normal subgroups only

This group has 428 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) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,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, 50447);