#352610

order	512
nilpotency class	3
isoclinism	isoclinic class of #348438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=fh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,c^b=chi,d^b=dh,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259806);