#474243

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

632 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)(9,10,14,19,15,20,22,33)(11,21,17,12,24,26,31,16)(13,27,32,36,30,18,29,37)(23,28,35,25,39,40,38,34);(1,2,4,5)(3,6,7,8)(9,11,22,17,15,24,14,31)(10,16,33,21,20,12,19,26)(13,23,29,35,30,39,32,38)(18,34,36,28,27,25,37,40);(3,7)(6,8)(9,12)(10,17)(11,19)(13,28)(14,21)(15,16)(18,23)(20,31)(22,26)(24,33)(25,29)(27,39)(30,40)(32,34)(35,36)(37,38);(1,3,4,7)(2,6,5,8)(9,13,15,30)(10,18,20,27)(11,23,24,39)(12,25,16,34)(14,32,22,29)(17,35,31,38)(19,37,33,36)(21,28,26,40)

Polycyclic presentation:

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

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, 381439);