#6437268

order	512
nilpotency class	3
isoclinism	isoclinic class of #29730
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344464);