#138711

order	512
nilpotency class	4
isoclinism	isoclinic class of #138689
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=i,d^2=ghi,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,h^c=hi

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, 45907);