#179365

order	512
nilpotency class	3
isoclinism	isoclinic class of #179229
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

843 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86561);