#501703

order	512
nilpotency class	4
isoclinism	isoclinic class of #501701
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24780
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24809	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408899);