#164957

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

927 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72153);