#168664

order	512
nilpotency class	3
isoclinism	isoclinic class of #168497
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

729 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75860);