#350261

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

815 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)(9,10,14,11)(12,19,21,24)(13,18,23,16)(15,22,17,20)(25,26,29,34)(27,32,30,31)(28,36,37,39)(33,38,40,35);(2,5)(6,8)(9,11,14,10)(12,17,21,15)(13,18,23,16)(19,20,24,22)(25,27,29,30)(26,31,34,32)(28,35,37,38)(33,36,40,39);(1,3,4,7)(2,6,5,8)(9,12,14,21)(10,15,11,17)(13,22,23,20)(16,24,18,19)(26,32)(27,30)(31,34)(33,39)(35,38)(36,40);(13,23)(16,18)(19,24)(20,22)(25,28,29,37)(26,33,34,40)(27,35,30,38)(31,36,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=eh,d^2=i,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, 257457);