#473865

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381061);