#164371

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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

751 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^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=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, 71567);