#266767

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
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

732 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)(9,10,13,17)(11,14,19,24)(12,15,21,16)(18,22,20,23)(25,26,29,32)(27,30,34,38)(28,31,35,39)(33,37,36,40);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,14,17,24)(12,20,21,18)(15,23,16,22)(27,33)(28,35)(30,37)(31,39)(34,36)(38,40);(10,15)(12,21)(14,22)(16,17)(18,20)(23,24)(25,27,29,34)(26,30,32,38)(28,36,35,33)(31,40,39,37);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

2^9|a^2=ehi,b^2=eh,c^2=i,d^2=e,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,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 173963);