#119517

order	512
nilpotency class	3
isoclinism	isoclinic class of #119510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

603 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fh,g^b=gh,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26713);