#440782

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 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)(11,21,24,16)(12,26,23,17)(13,28)(14,19)(15,31)(18,29)(20,30)(22,39)(25,35,32,36)(27,34,40,37)(33,38);(1,2,4,5)(3,6,7,8)(9,11,14,24)(10,16,19,21)(12,13,23,22)(15,32,30,25)(17,28,26,39)(18,37,38,34)(20,36,31,35)(27,33,40,29);(1,3,4,7)(2,6,5,8)(9,12,14,23)(10,17)(11,22,24,13)(15,27,30,40)(16,28)(18,36)(19,26)(20,34)(21,39)(25,33,32,29)(31,37)(35,38);(2,5)(6,8)(9,13,14,22)(10,18,19,38)(11,23,24,12)(15,33,30,29)(16,34,21,37)(17,36,26,35)(20,39,31,28)(25,27,32,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=ei,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, 347978);