#127320

order	512
nilpotency class	3
isoclinism	isoclinic class of #127312
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×H(3,ℤ/4)
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	C₂×#1289
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ghi,e^2=i,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=ci,e^b=eh,e^d=ei

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, 34516);