#350263

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

836 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)(9,10,14,12)(11,17,20,24)(13,21,23,16)(15,22,18,19)(25,26)(27,30)(28,31)(29,33)(32,36)(34,37)(35,38)(39,40);(2,5)(6,8)(25,27,29,34)(26,30,33,37)(28,35,36,40)(31,38,32,39);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,22)(14,20)(16,24)(17,21)(19,23)(27,34)(30,37)(35,40)(38,39);(9,12,14,10)(11,18,20,15)(13,16,23,21)(17,22,24,19)(26,31)(28,36)(30,38)(32,33)(35,40)(37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,c^b=ci,f^c=fh,g^c=gh,f^d=fh,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257459);