#266516

order	512
nilpotency class	3
isoclinism	isoclinic class of #247367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=i,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,d^b=dh,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, 173712);