#501936

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24967
prime radical	#28249	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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)(3,15)(4,21)(5,27)(6,13)(7,31)(8,14)(9,18)(10,25)(11,28)(12,30)(16,47)(17,52)(19,38)(20,36)(22,39)(23,40)(24,43)(26,41)(29,44)(32,37)(33,61)(34,51)(35,54)(42,59)(45,50)(46,53)(48,57)(49,56)(55,62)(58,64)(60,63);(1,3,8,20)(2,9)(4,22,26,58)(5,17,29,54)(6,18,30,19)(7,32,28,53)(10,39)(11,35)(12,15)(13,36)(14,38)(16,48,51,23)(21,56)(24,49,25,60)(27,46)(31,52)(33,55,59,50)(34,42)(37,44)(40,45)(41,63)(43,64)(47,61)(57,62);(1,4)(2,10)(3,16)(5,23)(6,24)(7,33)(8,26)(9,34)(11,40)(12,41)(13,21)(14,43)(15,45)(17,49)(18,50)(19,55)(20,51)(22,32)(25,30)(27,61)(28,59)(29,48)(31,57)(35,39)(36,62)(37,56)(38,47)(42,44)(46,63)(52,64)(53,58)(54,60);(1,5,6,28,8,29,30,7)(2,11,12,27,14,31,13,44)(3,17,18,53,20,54,19,32)(4,23,24,59,26,48,25,33)(9,35,15,46,38,52,36,37)(10,40,41,61,43,57,21,42)(16,49,50,58,51,60,55,22)(34,39,45,63,47,64,62,56)

Polycyclic presentation:

2^9|b^2=fh,d^2=e,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=gh,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, 409132);