#403855

order	512
nilpotency class	3
isoclinism	isoclinic class of #394813
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14883
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1107 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)(3,15,19,34)(4,21,25,10)(5,26,28,39)(6,12)(7,14)(8,13)(9,18,33,20)(11,27,37,30)(16,40)(17,44)(22,51)(23,53)(24,38,29,36)(31,47)(32,46)(35,56)(41,48)(42,57,52,63)(43,49)(45,58)(50,54)(55,59)(60,62,64,61);(1,3,8,20,7,19,6,18)(2,9,12,15,14,33,13,34)(4,22,29,43,25,16,24,47)(5,17,30,48,28,46,27,45)(10,31,36,40,21,49,38,51)(11,32,26,41,37,44,39,58)(23,52,56,61,55,42,54,62)(35,57,53,60,50,63,59,64);(1,4,7,25)(2,10)(3,16,19,22)(5,23,28,55)(6,24,8,29)(9,31)(11,35)(12,36)(13,38)(14,21)(15,40)(17,42,46,52)(18,43,20,47)(26,53)(27,54,30,56)(32,57)(33,49)(34,51)(37,50)(39,59)(41,60)(44,63)(45,61,48,62)(58,64);(1,5,6,27,7,28,8,30)(2,11,12,26,14,37,13,39)(3,17,18,45,19,46,20,48)(4,23,24,54,25,55,29,56)(9,32,15,41,33,44,34,58)(10,35,36,53,21,50,38,59)(16,42,43,61,22,52,47,62)(31,57,40,60,49,63,51,64)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311051);