#349225

order	512
nilpotency class	3
isoclinism	isoclinic class of #346294
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×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1060 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)(9,10,13,15)(11,18,20,24)(12,17,14,21)(16,22,23,19)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,39,40,36);(1,3,4,8)(2,5,6,7)(10,15)(16,23)(17,21)(18,24);(9,11)(10,16)(12,19)(13,20)(14,22)(15,23)(17,24)(18,21)(25,27)(26,30)(28,33)(29,34)(31,36)(32,37)(35,40)(38,39);(9,12,13,14)(10,17,15,21)(11,19,20,22)(16,24,23,18)(25,28,29,35)(26,31,32,38)(27,33,34,40)(30,36,37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256421);