#134725

order	512
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂×#41	M₁₆
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

329 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41921);