#474242

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

631 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)(9,10,16,20,15,22,14,21)(11,23,25,17,27,40,26,36)(12,28,29,38,30,37,34,18)(13,31,32,35,33,19,24,39);(1,2,4,5)(3,6,7,8)(9,11,16,25,15,27,14,26)(10,17,20,40,22,36,21,23)(12,24,34,33,30,32,29,13)(18,35,37,31,38,39,28,19);(1,3,4,7)(2,6,5,8)(9,12,14,29,15,30,16,34)(10,18,20,37,22,38,21,28)(11,24,25,33,27,32,26,13)(17,35,23,19,36,39,40,31);(3,7)(6,8)(9,13,14,32,15,33,16,24)(10,19,21,39,22,31,20,35)(11,12,26,29,27,30,25,34)(17,18,23,28,36,38,40,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=e,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,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, 381438);