#351329

order	512
nilpotency class	3
isoclinism	isoclinic class of #347668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

594 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)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,23,24,22)(26,31)(27,29)(28,36)(32,34)(33,40)(35,37);(1,3,4,7)(2,5,6,8)(25,26,27,32,30,34,29,31)(28,33,35,39,37,40,36,38);(2,6)(5,8)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,22,17,23)(25,27,30,29)(26,32,34,31)(28,36,37,35)(33,38,40,39);(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,28,29,36,30,37,27,35)(26,33,31,38,34,40,32,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=efhi,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, 258525);