#503181

order	512
nilpotency class	4
isoclinism	isoclinic class of #503179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25757
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61105	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1096 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=ei,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,g^c=gi

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, 410377);