#437710

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1126 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fi,c^2=efi,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,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, 344906);