#351633

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,20)(18,21)(19,22)(23,24)(25,26,31,34)(27,35,38,39)(28,29,30,33)(32,37,40,36);(1,3,4,7)(2,5,6,8)(10,14)(11,18)(16,17)(19,24)(25,27,30,37,31,38,28,36)(26,32,29,39,34,40,33,35);(2,6)(5,8)(9,11,12,18)(10,15,14,21)(13,19,20,24)(16,22,17,23)(25,28,31,30)(26,29,34,33)(27,36,38,37)(32,39,40,35);(10,16)(13,20)(14,17)(15,22)(19,24)(21,23)(25,29,30,34,31,33,28,26)(27,35,37,32,38,39,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=efi,c^2=i,d^2=ei,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=chi,d^b=dh,d^c=dh,g^d=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258829);