#110720

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

718 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,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, 17916);