#164293

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

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

756 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:

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

Polycyclic presentation:

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

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, 71489);