#466628

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

891 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,26)(13,29)(14,20)(15,23)(16,21)(17,18)(19,31)(24,34)(25,28)(27,39)(30,40)(32,36)(33,38)(35,37);(2,5)(6,8)(9,11,16,26,14,18,15,25)(10,17,21,28,20,22,23,12)(13,24,33,40,31,35,32,39)(19,34,36,30,29,37,38,27);(1,3)(2,6)(4,7)(5,8)(9,12,15,17,14,28,16,22)(10,18,21,26,20,11,23,25)(13,30,32,37,31,27,33,34)(19,24,36,39,29,35,38,40);(3,7)(6,8)(9,13,15,32,14,31,16,33)(10,19,21,36,20,29,23,38)(11,24,25,39,18,35,26,40)(12,27,17,34,28,30,22,37)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,g^c=gi,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, 373824);