#349362

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
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	C₂³×C₄	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh,g^c=gi,f^d=fi

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, 256558);