#168031

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

949 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)(10,14)(12,19)(15,21)(16,20)(18,23)(22,24)(25,26)(27,35)(28,33)(29,32)(30,34)(31,37)(36,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,15,17,22)(12,20,19,14)(18,24,23,21)(25,27,30,38)(26,31,34,40)(28,36,29,37)(32,39,33,35);(1,3,4,7)(2,5,6,8)(9,11,13,17)(10,15,16,22)(12,18,19,23)(14,21,20,24)(26,32)(28,29)(31,39)(33,34)(35,40)(36,37);(11,17)(15,22)(18,23)(21,24)(25,28,30,29)(26,33,34,32)(27,36,38,37)(31,35,40,39)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=i,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, 75227);