#164411

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

746 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)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19)(25,26,29,34,30,32,28,33)(27,35,37,31,38,40,36,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18)(25,27,30,38)(26,31,32,39)(28,36,29,37)(33,35,34,40);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,21)(13,16)(14,17)(18,19)(20,24)(22,23)(26,32)(31,39)(33,34)(35,40);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,28,30,29)(26,33,32,34)(27,36,38,37)(31,35,39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,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, 71607);