#266282

order	512
nilpotency class	3
isoclinism	isoclinic class of #247207
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

783 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ehi,c^2=hi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=dh,f^c=fh,g^c=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, 173478);