#486332

order	512
nilpotency class	4
isoclinism	isoclinic class of #486327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#62279	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

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, 393528);