#440774

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
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	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1150 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)(9,10,13,27,14,18,30,19)(11,20,22,16,23,15,28,33)(12,24,25,35,26,17,29,36)(21,34,37,40,38,31,39,32);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,21)(13,28)(14,23)(16,27)(17,34)(18,20)(19,33)(22,30)(24,31)(25,39)(26,38)(29,37)(32,36)(35,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16,18,33)(12,21)(13,22)(14,23)(15,19,20,27)(17,32,24,40)(25,37)(26,38)(28,30)(29,39)(31,36,34,35);(2,5)(6,8)(9,12,14,26)(10,17,18,24)(11,21,23,38)(13,29,30,25)(15,31,20,34)(16,32,33,40)(19,36,27,35)(22,39,28,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,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, 347970);