#126713

order	512
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

405 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:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(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)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(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);(3,9)(6,12)(10,15)(13,16)(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,d^2=g,e^2=h,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=chi,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33909);