#252218

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59303	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

923 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,18,24)(12,20,21,15)(14,22,23,19)(25,26,28,30)(27,31,34,37)(29,32,35,38)(33,40,36,39);(1,3,4,8)(2,5,6,7)(12,21)(14,23)(15,20)(19,22)(26,30)(27,33)(29,35)(31,39)(34,36)(37,40);(9,11)(10,14)(12,22)(13,18)(15,24)(16,23)(17,20)(19,21)(25,27,28,34)(26,31,30,37)(29,36,35,33)(32,39,38,40);(2,6)(5,7)(11,18)(14,23)(17,24)(19,22)(26,30)(31,37)(32,38)(39,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,f^c=fh,g^c=gi

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, 159414);