#451782

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1280 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)(11,15)(12,14)(13,16)(17,18,20,22)(19,21,23,24)(25,26,30,31,29,34,27,33)(28,36,38,40,37,32,35,39);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,23)(22,24)(25,27,29,30)(26,31,34,33)(28,35,37,38)(32,39,36,40);(10,14)(13,15)(19,23)(21,24)(25,28,29,37)(26,32,34,36)(27,35,30,38)(31,39,33,40);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(26,33)(27,30)(31,34)(32,40)(35,38)(36,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^d=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, 358978);