#500736

order	512
nilpotency class	4
isoclinism	isoclinic class of #500736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24236
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1016 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=e,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407932);