#168608

order	512
nilpotency class	3
isoclinism	isoclinic class of #168497
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

813 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=h,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, 75804);