#148374

order	512
nilpotency class	4
isoclinism	isoclinic class of #135932
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#442
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

692 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,g^b=gi,d^c=dhi,f^c=fhi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55570);