#352630

order	512
nilpotency class	3
isoclinism	isoclinic class of #352047
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

665 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)(9,10,14,16)(11,17,18,24)(12,13,20,21)(15,22,23,19)(25,26)(27,33)(28,36)(29,31)(30,34)(32,37)(35,39)(38,40);(2,5)(6,8)(9,11,13,19,14,18,21,22)(10,15,12,17,16,23,20,24);(11,18)(15,23)(17,24)(19,22)(25,27,28,34,29,35,37,40)(26,30,32,33,31,38,36,39);(1,3,4,7)(2,6,5,8)(9,12,13,16,14,20,21,10)(11,17,19,23,18,24,22,15)(26,31)(30,38)(32,36)(33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259826);