#501261

order	512
nilpotency class	4
isoclinism	isoclinic class of #501260
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24424
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61100	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#444 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1026 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408457);