#167008

order	512
nilpotency class	3
isoclinism	isoclinic class of #166993
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

957 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)(3,5)(4,7)(6,8)(9,10,14,18)(11,19,22,23)(12,17,13,16)(15,20,24,21)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,39,40,36);(9,11,14,22)(10,15,18,24)(12,21,13,20)(16,19,17,23)(25,27)(26,30)(28,33)(29,34)(31,36)(32,37)(35,40)(38,39);(2,5)(3,8)(6,7)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,40)(37,39);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 74204);