#463365

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

770 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=f,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=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, 370561);