#290917

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

736 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,14)(13,21)(15,20)(16,19)(18,23)(22,24)(28,36)(30,38)(32,37)(35,40);(1,3,4,8)(2,5,6,7)(10,14)(11,18)(13,19)(15,22)(25,26)(27,33)(28,37)(29,31)(30,35)(32,36)(34,39)(38,40);(9,11,19,24,12,18,13,20)(10,15,21,23,14,22,16,17)(25,27,29,34)(26,30)(28,35,36,40)(31,38)(32,33)(37,39);(2,6)(5,7)(10,14)(15,22)(16,21)(17,23)(27,34)(30,38)(33,39)(35,40)

Polycyclic presentation:

2^9|b^2=eh,c^2=fi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198113);