#512253

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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)(3,5)(4,6)(7,8)(10,17)(11,23)(12,26)(13,15)(14,25)(18,33)(19,35)(21,22)(24,29)(27,39)(28,30)(32,37)(34,36)(38,40);(1,3,4,7)(2,5,6,8)(9,10,13,17,16,22,15,21)(11,18,23,31,25,33,14,20)(12,19,27,32,29,38,28,36)(24,34,39,40,26,37,30,35);(2,6)(5,8)(9,11,13,23,16,25,15,14)(10,18,17,31,22,33,21,20)(12,24,27,39,29,26,28,30)(19,34,32,40,38,37,36,35);(9,12)(10,19)(11,24)(13,27)(14,30)(15,28)(16,29)(17,32)(18,34)(20,35)(21,36)(22,38)(23,39)(25,26)(31,40)(33,37)

Polycyclic presentation:

2^9|b^2=efh,c^2=e,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,g^c=gi,f^d=fi,g^d=gh,h^d=hi

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, 419449);