#409361

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

952 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,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, 316557);