#141846

order	512
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#476
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9864
prime radical	C₂³	#421
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 49042);