#339905

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

697 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:

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

Polycyclic presentation:

2^9|b^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, 247101);