#473866

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1222 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)(25,26,28,33,29,32,36,31)(27,30,35,38,34,39,40,37);(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)(26,30)(28,35)(29,34)(31,37)(32,39)(33,38)(36,40);(1,3)(2,6)(4,7)(5,8)(11,15)(13,16)(26,31)(28,36)(30,37)(32,33)(35,40)(38,39);(3,7)(6,8)(18,22)(21,24)(26,32)(27,34)(31,33)(35,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=eghi,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, 381062);