#469904

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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 ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1022 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377100);