#446375

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

980 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efi,c^2=fgi,d^2=f,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, 353571);