#141764

order	512
nilpotency class	4
isoclinism	isoclinic class of #141746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

303 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=egi,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48960);