#352784

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

732 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)(10,14)(11,18)(12,15)(16,21)(17,19)(20,23)(27,33)(28,35)(30,36)(32,38)(34,39)(37,40);(2,5)(6,8)(9,10,15,21,13,16,12,14)(11,19,23,22,20,17,18,24)(25,26,29,31)(27,30,34,37)(28,32,35,38)(33,36,39,40);(10,16)(14,21)(17,19)(22,24)(25,27,28,33,29,34,35,39)(26,30,32,36,31,37,38,40);(1,3,4,7)(2,6,5,8)(9,11,12,18,13,20,15,23)(10,17,14,22,16,19,21,24)(26,31)(30,37)(32,38)(36,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=f,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=dhi

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, 259980);