#141730

order	512
nilpotency class	4
isoclinism	isoclinic class of #141715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 102 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)(3,7)(4,8)(5,6)(9,10,14,21,15,19,16,37)(11,17,25,40,23,41,36,42)(12,26,30,59,31,57,32,64)(13,33,28,48,34,45,35,20)(18,43,47,62,46,63,58,29)(22,49,52,71,27,56,61,72)(24,54,55,39,53,66,50,69)(38,65,67,60,44,70,68,51);(1,3,7,2,5,8,4,6)(9,11,16,25,15,23,14,36)(10,17,21,42,19,41,37,40)(12,27,32,61,31,22,30,52)(13,24,35,55,34,53,28,50)(18,44,47,67,46,38,58,68)(20,39,33,54,48,69,45,66)(26,56,59,71,57,49,64,72)(29,60,63,70,62,51,43,65);(2,6)(3,8)(9,12,28,43,16,32,13,29,15,31,35,63,14,30,34,62)(10,18,45,64,21,47,20,26,19,46,33,59,37,58,48,57)(11,22,50,70,25,52,24,51,23,27,55,65,36,61,53,60)(17,38,66,71,42,68,39,49,41,44,54,72,40,67,69,56)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egh,d^2=gi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,d^b=di,f^b=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48926);