#474958

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

1038 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,d^2=gi,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, 382154);