#512215

order	512
nilpotency class	4
isoclinism	isoclinic class of #511963
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	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 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)(11,13)(12,26)(14,19)(15,21)(16,20)(17,22)(18,29)(23,28)(24,31)(25,32)(27,37)(30,36)(33,35)(34,39)(38,40);(2,5)(6,8)(9,10,14,11,16,21,32,24,17,22,31,25,15,20,13,19)(12,18,28,23,29,26,39,37,30,35,40,38,33,36,27,34);(1,3,4,7)(2,6,5,8)(9,11,13,10,15,19,31,20,17,25,32,22,16,24,14,21)(12,23,27,18,33,34,40,36,30,38,39,35,29,37,28,26);(9,12)(10,18)(11,23)(13,27)(14,28)(15,33)(16,29)(17,30)(19,34)(20,36)(21,26)(22,35)(24,37)(25,38)(31,40)(32,39)

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,c^b=ci,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419411);