#474792

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
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

620 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)(3,5)(4,6)(7,8)(9,10,15,20)(11,21)(12,26,27,34)(13,16,14,19)(17,23)(18,29,36,30)(22,32)(24,33)(25,31)(28,40)(35,38)(37,39);(1,3,4,7)(2,5,6,8)(10,16,20,19)(12,27)(17,31,33,21)(18,34,36,26)(22,37)(28,38)(29,30)(32,35,39,40);(9,11)(10,17,20,33)(12,28)(13,24)(14,23)(15,25)(16,21,19,31)(18,35,36,40)(22,29)(26,39,34,32)(27,38)(30,37);(2,6)(5,8)(9,12,13,29,15,27,14,30)(10,18,19,34,20,36,16,26)(11,22,23,28,25,37,24,38)(17,32,21,35,33,39,31,40)

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,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, 381988);