#339913

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,f^c=fi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247109);