#333666

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

734 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,f^c=fi,e^d=eh,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, 240862);