#351323

order	512
nilpotency class	3
isoclinism	isoclinic class of #347668
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

586 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)(11,14)(12,15)(13,17)(16,20)(18,21)(19,23)(22,24)(25,26,31,34)(27,35,39,37)(28,33,30,29)(32,36,40,38);(1,3,4,7)(2,5,6,8)(25,27,28,36,31,39,30,38)(26,32,29,37,34,40,33,35);(2,6)(5,8)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22)(25,28,31,30)(26,33,34,29)(27,36,39,38)(32,35,40,37);(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)(25,29,30,26,31,33,28,34)(27,37,38,32,39,35,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258519);