#137872

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,8)(3,6)(4,9,13,14)(7,12)(10,16)(11,15)(17,18,20,24,21,25,28,31,22,19,23,26,29,27,30,32);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,19,20,26,21,27,28,32,22,18,23,24,29,25,30,31);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45068);