#164393

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

792 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)(11,18)(12,14)(13,17)(15,19)(16,21)(20,23)(22,24)(25,26,29,32,30,33,28,37)(27,34,35,39,36,31,38,40);(1,2)(3,6)(4,8)(5,7)(10,14)(12,21)(15,22)(16,17)(18,23)(20,24)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,34)(37,40);(9,11,13,19)(10,15,17,18)(12,20,21,24)(14,22,16,23)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,34,40);(1,3,5,4)(2,6,7,8)(11,19)(15,18)(20,24)(22,23)

Polycyclic presentation:

2^9|a^2=f,c^2=g,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=dgi

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, 71589);