#169494

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
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	#39444	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 76690);