#476354

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1038 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383550);