#349315

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
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	C₂×#562	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

875 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,c^b=ch,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256511);