#478370

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
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

808 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(18,20)(19,23)(25,26,28,30)(27,34,35,31)(29,32,37,33)(36,38,40,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,20)(19,24,23,22)(26,30)(31,34)(32,33)(38,39);(2,6)(5,8)(11,15)(13,16)(17,19,21,23)(18,22,20,24)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40);(3,7)(5,8)(17,20,21,18)(19,22,23,24)(26,32)(27,35)(29,37)(30,33)(31,38)(34,39)

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=ei,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,h^d=hi

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, 385566);