#440778

order	512
nilpotency class	3
isoclinism	isoclinic class of #419459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=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, 347974);