#476352

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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1007 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 383548);