#475222

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = 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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(25,26,30,34)(27,35,37,32)(28,31,36,40)(29,39,38,33);(2,5)(6,8)(17,18)(19,22)(20,21)(23,24)(25,27,29,28,30,37,38,36)(26,31,33,35,34,40,39,32);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(18,21)(22,24)(27,36)(28,37)(29,38)(31,32)(33,39)(35,40);(3,7)(6,8)(17,19,20,23)(18,22,21,24)(25,28,38,27,30,36,29,37)(26,32,39,40,34,35,33,31)

Polycyclic presentation:

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

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, 382418);