#504266

order	512
nilpotency class	4
isoclinism	isoclinic class of #504265
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26464
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=fh,d^2=ei,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=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, 411462);