#440780

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
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

1166 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)(11,16,21,15)(12,23)(13,26)(14,18)(17,24)(19,29)(20,36,37,40)(22,31,27,33)(25,35)(28,34)(30,38,32,39);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,20)(13,27)(14,21)(16,18)(17,32)(19,33)(22,29)(23,40)(24,39)(25,37)(26,31)(28,38)(30,34)(35,36);(1,3,4,7)(2,6,5,8)(9,11)(10,16,18,15)(12,20)(13,22)(14,21)(17,32,34,30)(19,33,26,31)(23,36,35,40)(24,38)(25,37)(27,29)(28,39);(2,5)(6,8)(9,12,14,25)(10,17,18,34)(11,20,21,37)(13,28,29,24)(15,30,16,32)(19,35,26,23)(22,39,27,38)(31,36,33,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=e,d^2=i,h^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,f^c=fi,h^d=hi

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, 347976);