#512212

order	512
nilpotency class	4
isoclinism	isoclinic class of #511960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

616 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)(10,17)(11,19)(12,23)(13,28)(14,30)(15,29)(18,21)(20,33)(22,32)(25,40)(26,27)(31,34)(35,37)(36,38);(2,5)(6,8)(9,10,14,11,15,19,13,17,16,20,28,22,29,32,30,33)(12,18,25,21,26,35,24,31,27,36,40,38,23,34,39,37);(1,3,4,7)(2,6,5,8)(9,11,13,20,29,33,14,19,16,22,30,10,15,17,28,32)(12,21,24,36,23,37,25,35,27,38,39,18,26,31,40,34);(9,12)(10,18)(11,21)(13,24)(14,25)(15,26)(16,27)(17,31)(19,35)(20,36)(22,38)(23,29)(28,40)(30,39)(32,34)(33,37)

Polycyclic presentation:

2^9|b^2=f,c^2=e,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419408);