#501856

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	#24897
prime radical	#33538	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

631 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)(5,26)(6,13)(7,12)(8,14)(9,17)(10,24)(11,30)(16,48)(18,41)(19,46)(20,47)(22,57)(23,42)(25,39)(27,34)(28,38)(29,44)(31,43)(32,40)(33,58)(35,53)(36,54)(37,52)(45,50)(49,55)(51,64)(56,60)(59,63)(61,62);(1,3)(2,9)(4,22,25,50)(5,27)(6,19)(7,18)(8,20)(10,37,40,55)(11,41)(12,35)(13,34)(14,36)(15,26)(16,24,51,32)(17,30)(21,33,39,63)(23,49,28,52)(29,54)(31,53)(38,45,42,57)(43,47)(44,46)(48,56,64,61)(58,62,59,60);(1,4)(2,10)(3,16)(5,28)(6,32)(7,24)(8,25)(9,33)(11,42)(12,39)(13,21)(14,40)(15,45)(17,52)(18,22)(19,50)(20,51)(23,31)(26,61)(27,58)(29,60)(30,62)(34,55)(35,37)(36,63)(38,44)(41,48)(43,56)(46,64)(47,57)(49,54)(53,59);(1,3)(2,9)(4,16)(5,27)(6,19)(7,18)(8,20)(10,33)(11,41)(12,35)(13,34)(14,36)(15,26)(17,30)(21,55)(22,24)(23,59)(25,51)(28,58)(29,54)(31,53)(32,50)(37,39)(38,64)(40,63)(42,48)(43,47)(44,46)(45,61)(49,60)(52,62)(56,57)

Polycyclic presentation:

2^9|b^2=fh,e^2=hi,f^2=i,g^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,e^b=eh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=fi,h^d=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, 409052);