#127965

order	512
nilpotency class	4
isoclinism	isoclinic class of #7262
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=hi,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ci,d^b=dh,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, 35161);