#470509

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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	#2684
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

654 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=hi

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, 377705);