#131409

order	512
nilpotency class	4
isoclinism	isoclinic class of #131330
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	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 622 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)(9,10,14,20,34,48,35,49,16,22,13,19,15,21,37,51)(11,23,27,42,38,50,32,17,26,41,25,52,28,46,36,40)(12,18,29,43,60,68,64,72,31,45,33,47,30,44,61,69)(24,54,55,70,62,71,63,65,56,53,58,66,57,67,59,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,24)(13,32)(14,36)(15,38)(16,26)(18,39)(19,46)(20,50)(21,52)(22,40)(23,49)(25,35)(27,37)(28,34)(29,59)(30,62)(31,56)(33,63)(41,51)(42,48)(43,67)(44,70)(45,65)(47,71)(53,72)(54,69)(55,61)(57,60)(58,64)(66,68);(1,4,5,6)(2,3,7,8)(9,12)(10,18)(11,24)(13,33)(14,29)(15,30)(16,31)(17,39)(19,47)(20,43)(21,44)(22,45)(23,53)(25,58)(26,56)(27,55)(28,57)(32,63)(34,60)(35,64)(36,59)(37,61)(38,62)(40,65)(41,54)(42,66)(46,71)(48,68)(49,72)(50,67)(51,69)(52,70)

Polycyclic presentation:

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

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, 38605);