#352789

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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

673 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)(9,10,14,15)(11,18,17,13)(12,20,22,19)(16,24,23,21)(27,33)(28,34)(30,36)(32,37)(35,39)(38,40);(2,5)(6,8)(9,11,13,10,14,17,18,15)(12,20,21,23,22,19,24,16)(25,26)(27,30)(28,32)(29,31)(33,36)(34,37)(35,38)(39,40);(10,15)(11,17)(16,23)(19,20)(25,27,34,39,29,35,28,33)(26,30,37,40,31,38,32,36);(1,3,4,7)(2,6,5,8)(9,12,13,21,14,22,18,24)(10,16,17,20,15,23,11,19)(26,31)(30,38)(32,37)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fhi,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, 259985);