#501998

order	512
nilpotency class	4
isoclinism	isoclinic class of #501996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25032
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=ehi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409194);