#475713

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=egh,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,f^c=fi

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, 382909);