#305863

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

565 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)(9,10,14,16)(11,17,20,24)(12,21,19,15)(13,23,22,18)(25,26,28,31)(27,33,35,30)(29,32,37,39)(34,38,36,40);(9,11,13,19,14,20,22,12)(10,15,18,17,16,21,23,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,40)(36,37)(38,39);(1,3,4,8)(2,5,6,7)(10,16)(15,21)(17,24)(18,23)(26,31)(27,34)(29,37)(30,38)(33,40)(35,36);(9,12,22,20,14,19,13,11)(10,17,23,15,16,24,18,21)(26,31)(30,33)(32,39)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=fhi,d^2=ehi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 213059);