#164534

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
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	#36360	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1071 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 71730);