#110144

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

716 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh

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, 17340);