#170342

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#39445	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77538);