#6436335

order	512
nilpotency class	3
isoclinism	isoclinic class of #29730
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1518 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,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, 6343531);