#499779

order	512
nilpotency class	4
isoclinism	isoclinic class of #499777
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23848
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1245 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)(5,17)(6,29)(7,13)(8,12)(9,14)(10,25)(11,33)(16,45)(18,51)(19,31)(20,44)(22,54)(23,56)(24,58)(26,39)(27,49)(28,50)(30,36)(32,42)(34,41)(35,40)(37,46)(38,62)(43,47)(48,60)(52,55)(53,61)(57,59)(63,64);(1,3)(2,5)(4,22)(6,30)(7,18)(8,19)(9,20)(10,37)(11,31)(12,27)(13,36)(14,28)(15,29)(16,25)(17,33)(21,52)(23,39)(24,59)(26,47)(32,50)(34,49)(35,48)(38,43)(40,57)(41,44)(42,51)(45,53)(46,62)(54,58)(55,61)(56,63)(60,64);(1,4,9,26)(2,10,14,40)(3,16,20,48)(5,23,28,52)(6,24,34,62)(7,35,8,25)(11,38,42,58)(12,39,13,21)(15,43,44,54)(17,46,50,59)(18,22,19,47)(27,57,36,37)(29,53,41,64)(30,55,49,56)(31,60,51,45)(32,63,33,61);(1,5)(2,3)(4,23)(6,31)(7,27)(8,36)(9,28)(10,16)(11,30)(12,18)(13,19)(14,20)(15,33)(17,29)(21,47)(22,39)(24,60)(25,37)(26,52)(32,44)(34,51)(35,57)(38,55)(40,48)(41,50)(42,49)(43,61)(45,62)(46,53)(54,63)(56,58)(59,64)

Polycyclic presentation:

2^9|c^2=i,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,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406975);