#131373

order	512
nilpotency class	4
isoclinism	isoclinic class of #131284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

611 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,4,8,6)(5,7)(9,10,15,21,37,20,38,53,16,23,13,19,14,22,40,52)(11,24,28,45,39,55,35,17,29,44,26,43,27,50,41,46)(12,30,31,64,32,18,47,48,34,54,36,68,33,49,66,51)(25,57,58,71,59,56,67,70,61,72,62,42,60,69,65,63);(1,3)(2,6)(4,7)(5,8)(9,11,16,29)(10,17,23,46)(12,25,34,61)(13,35,15,41)(14,39,37,27)(18,42,49,71)(19,50,21,55)(20,45,22,43)(24,52,44,53)(26,40,28,38)(30,63,54,70)(31,65,36,67)(32,60,33,59)(47,62,66,58)(48,72,51,57)(56,64,69,68);(1,4,5,6)(2,3,7,8)(9,12,16,34)(10,18,23,49)(11,25,29,61)(13,36,15,31)(14,33,37,32)(17,42,46,71)(19,51,21,48)(20,54,22,30)(24,56,44,69)(26,62,28,58)(27,60,39,59)(35,67,41,65)(38,47,40,66)(43,63,45,70)(50,57,55,72)(52,64,53,68)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei,f^c=fg

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, 38569);