#510491

order	512
nilpotency class	4
isoclinism	isoclinic class of #510490
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31665
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1773 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

444 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)(3,15,20,35)(4,21)(5,12)(6,10)(7,13)(8,14)(9,18,37,28)(11,23)(16,36,30,33)(17,34,50,60)(19,44,51,59)(22,39,48,61)(24,53)(25,42)(26,40)(27,43)(29,38)(31,57)(32,45)(41,54)(46,55)(47,64,56,63)(49,58,52,62);(1,3,6,28,8,20,5,18)(2,9,12,15,14,37,10,35)(4,22,23,50,26,48,25,17)(7,30,29,19,32,16,27,51)(11,39,21,34,42,61,40,60)(13,44,43,33,45,59,38,36)(24,52,54,56,31,49,55,47)(41,62,53,64,46,58,57,63);(2,10)(3,16,20,30)(5,6)(7,29)(9,33,37,36)(11,21)(12,14)(13,45)(15,44,35,59)(17,47,50,56)(18,51,28,19)(22,49,48,52)(23,25)(24,55)(27,32)(31,54)(34,58,60,62)(39,63,61,64)(40,42)(41,46);(1,4,8,26)(2,11,14,42)(3,17)(5,23,6,25)(7,31,32,24)(9,34)(10,21,12,40)(13,46,45,41)(15,39)(16,47)(18,22)(19,52)(20,50)(27,55,29,54)(28,48)(30,56)(33,58)(35,61)(36,62)(37,60)(38,57,43,53)(44,63)(49,51)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417687);