#479798

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

616 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386994);