#349373

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
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	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

1014 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)(11,15)(12,17)(13,22)(14,18)(16,19)(20,24)(21,23)(25,26,29,30)(27,31,34,36)(28,32,35,37)(33,39,40,38);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24)(26,30)(31,36)(32,37)(38,39);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,27)(26,31)(28,33)(29,34)(30,36)(32,38)(35,40)(37,39);(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)(25,28,29,35)(26,32,30,37)(27,33,34,40)(31,38,36,39)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=hi,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, 256569);