#331445

order	512
nilpotency class	3
isoclinism	isoclinic class of #331190
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

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

717 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)(9,10,13,17,12,16,19,23)(11,18,21,24,20,14,22,15)(26,30)(31,38)(32,39)(33,40);(2,5)(6,8)(9,11,19,22,12,20,13,21)(10,14,23,24,16,18,17,15)(25,26,29,32,28,30,36,39)(27,33,35,38,34,40,37,31);(3,7)(6,8)(9,11,12,20)(10,15,16,24)(13,22,19,21)(14,23,18,17)(25,27)(26,31)(28,34)(29,37)(30,38)(32,40)(33,39)(35,36);(1,3,4,7)(2,6,5,8)(11,20)(14,18)(15,24)(21,22)(27,34)(31,38)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=h,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

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, 238641);