#131669

order	512
nilpotency class	4
isoclinism	isoclinic class of #131668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62437	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

587 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=eg,f^c=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38865);