#167847

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

760 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)(10,14)(12,19)(15,21)(16,20)(18,23)(22,24)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,39,40,36);(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,15,17,22)(12,20,19,14)(18,24,23,21)(25,27,29,34)(26,30,32,37)(28,33,35,40)(31,36,38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,23)(20,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,40)(37,39);(11,17)(15,22)(18,23)(21,24)

Polycyclic presentation:

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

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, 75043);