#136660

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂³	#435
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(2,6,8,11)(3,9,10,12)(17,18,21,26,23,28,37,45,22,27,36,44,20,25,30,39)(19,29,33,46,32,42,47,43,34,40,35,48,31,24,38,41);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19,30,47,20,31,36,33,22,34,37,38,23,32,21,35)(18,24,39,46,25,40,44,41,27,42,45,48,28,29,26,43);(1,3,5,10)(2,7,8,16)(4,12,13,9)(6,14,11,15)(17,20,22,23)(18,25,27,28)(19,31,34,32)(21,30,36,37)(24,40,42,29)(26,39,44,45)(33,38,35,47)(41,48,43,46)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=egi,f^c=fgi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43856);