#430551

order	512
nilpotency class	3
isoclinism	isoclinic class of #417732
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

876 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,d^2=efi,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=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337747);