#127286

order	512
nilpotency class	4
isoclinism	isoclinic class of #127249
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34482);