#510423

order	512
nilpotency class	4
isoclinism	isoclinic class of #510422
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31601
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1773 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

381 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)(3,15,21,41)(4,22)(5,10)(6,13)(7,12)(8,14)(9,18,35,19)(11,25)(16,32,48,34)(17,49,51,33)(20,43,47,61)(23,55,52,37)(24,40)(26,54)(27,39)(28,36)(29,59)(30,42)(31,44)(38,58)(45,57)(46,64,56,63)(50,62,53,60);(1,3)(2,9,14,35)(4,23,27,52)(5,18)(6,16)(7,19)(8,21)(10,15,12,41)(11,37)(13,43,44,61)(17,24,51,25)(20,30)(22,49)(26,53,29,50)(28,47)(31,48)(32,42,34,36)(33,39)(38,62)(40,55)(45,60)(46,58,56,57)(54,63)(59,64);(2,10)(3,16,21,48)(5,7)(6,28)(9,32,35,34)(11,22)(12,14)(13,44)(15,43,41,61)(17,46,51,56)(18,20,19,47)(23,53,52,50)(24,25)(26,57)(29,58)(30,31)(33,60,49,62)(37,63,55,64)(38,45)(39,40);(1,4,8,27)(2,11,14,40)(3,17)(5,24,7,25)(6,29,31,26)(9,33)(10,22,12,39)(13,45,44,38)(15,37)(16,46)(18,23)(19,52)(20,53)(21,51)(28,58,30,57)(32,60)(34,62)(35,49)(36,59,42,54)(41,55)(43,63)(47,50)(48,56)(61,64)

Polycyclic presentation:

2^9|a^2=eh,b^2=fh,c^2=eh,d^2=ehi,e^2=i,f^2=i,g^2=hi,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=ghi,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, 417619);